SOME PROPERTIES OF TORSION CLASSES OF ABELIAN GROUPS
by
B.J. GARDNER, B.Sc. (Hons.)
Submitted in fulfilment of the requirements for
the degree of
Doctor of Philosophy
UNIVERSITY OF TASMANIA
HOBART
July, 1970
ACKNOWLEDGEMENTS
I wish to thank Dr. B.D. Jones for the guidance and
encouragement he has given during the time I have spent on this
research. I have borrowed his proof of a result concerning
torsionless abelian groups (PROPOSITION 2.29, p.31) in order to
dispose of a conjecture to which the reader might otherwise be
led by the adjacent discussion.
I am grateful also to Mrs. Judy Lindsay whose good
humour and efficiency have together helped to make the typing
and printing of the thesis a much less tedious affair than it
might have been.
B.J.G.
Except as stated herein, this thesis contains no
material which has been accepted for the award of any other
degree or diploma in any university, and to the best of my
knowledge and belief, contains no copy or paraphrase of
material previously published or written by another person
except where duly acknowledged.
eacto-c B.J. Gardner.
CONTENTS
INTRODUCTION (v)
TABLE OF NOTATION (ix)
CHAPTER 1. PRELIMINARIES 1
1. Torsion Theories 1 2. Torsion Classes and Radicals 3 3. The Kurosh Construction 7 4. The Amitsur Construction 9 5. Torsion Classes of Abelian Groups 11
CHAPTER 2. SOME TORSION CLASSES DETERMINED BY TORSION-FREE GROUPS 16
1. Rational Groups and Torsion Classes . 17 2. Further Examples 22 3. Some Large Torsion Classes 27
CHAPTER 3. ADDITIONAL CLOSURE PROPERTIES FOR TORSION CLASSES I: DIRECT PRODUCTS, PURE SUBGROUPS 33
1. Direct Products 33 2. Pure Subgroups 39
CHAPTER 4. ADDITIONAL CLOSURE PROPERTIES FOR TORSION CLASSES II: GENERALIZED PURE SUBGROUPS 47
1. Generalized Purity 48 2. Idempotence of Composite Radicals 53 3. A Simplification of the Problem 4. Generalized Rank 5. Groups of IL-rank 1 6. Groups of ilk-rank 1 (continued) 82 7. An Example 91
CHAPTER 5. MISCELLANEOUS TOPICS 95
1. Rational Groups and the Amitsur Construction 95 2. Splitting Idempotent Radicals 98
INDEX
190
REFERENCES 101
IN
The subject matter of this thesis has its origins in
Dickson's generalization to certain abelian categories of the
notion of torsion as applied to abelian groups [9] and the
earlier work of Kurosh [27] and Amitsur [1] on a general theory
of radicals for rings and algebras.
In [9] a torsion theory was defined as a couple (a,;..;)
of classes of objects such that
(i):7 n 5- = {0}
(ii):7 is closed under homomorphic images
(iii) is closed under subobjects
and (iv) for every object K there is a short exact
sequence
with T c :7 and F
In this situation (resp.a) was called a torsion (resp.
torsion-free) class. In the motivating example,a(resp. )
is the class of all torsion (resp. torsion-free) abelian groups.
Kurosh [27] defined a radical class of rings as a
class e such that
(i) C is closed under conormal epimorphic images
(ii) every ring A has an ideal CA belonging to e.
and containing all other such ideals
and (iii) C. (A/CA) = 0 for every ring A.
Kurosh's theory is applicable to quite general categories
(v)
("normal subobject" replaces "ideal" in (ii)), in particular to
the abelian categories considered in [9], for which the concepts
"torsion class" and "radical class" coincide. The function
which assigns CA to A becomes a functor if its action on
morphisms is defined by restriction. Such a functor is called
an idempotent radical. (Some authors omit "idempotent").
A class :T is a torsion class if and only if it is closed under homomorphic images, direct suns and extensions. In
considering specific torsion theories for modules, it has been
the practice of most authors to impose an extra condition:
theta be closed under submodules, or hereditary, and indeed
the analogy with abelian group torsion becomes a little tenuous
in the absence of this condition. One expects, for example,
that "torsion modules" be in some way characterized by the
annihilators of their elements. Nevertheless the language of
torsion classes and theories as described above, rather than
that of radical classes, has become standard, at least among
non-Russian authors writing about abelian categories.
In this thesis we are mainly concerned with torsion
classes of abelian groups. In parts of Chapter 4, however, we
work in abelian categories satisfying various sets of conditions,
as the added generality involves no significant complication
of the arguments.
Chapter 1 is essentially a catalogue of other people's
results which are referred to in the text. The relation between
Dickson's work and that of Kurosh and Amitsur is also briefly
discussed.
(vii)
The closure properties which characterize torsion
classes ensure that any group or class of groups is contained
in a smallest torsion class; this torsion class is said to be
determined by the group or class in question.
Some examples of torsion classes determined by torsion-
free groups are considered in Chapter 2. The behaviour of
rational groups as members of torsion classes is investigated
and the torsion classes determined by rational and torsion groups
are completely described.
The principal results are to be found in Chapters 3
and 4, where additional closure properties for torsion classes
are considered. It is shown in §1 of Chapter 3 that a torsion
class is closed under countable direct products, (i.e. direct
products of countable sets of groups) if and only if it is
determined by torsion-free groups. The remainder of the chapter
if devoted to closure under pure subgroups. The torsion classes
with this property are characterized - such a torsion class either
contains only torsion groups or is determined by a subring of
the rationals and a set of primary cyclic groups - and the result
generalized to obtain a description of those closed under S-pure
subgroups, where S is a set of primes.
In Chapter 4 we consider torsion classes closed under
certain generalized pure subgroups as defined by Carol Walker
in [41]. The following question is investigated: if 1A., anda
are torsion classes, when is a closed undera-pure subgroups? This question is actually a generalization of the one for ordinary
pure subgroups answered in Chapter 3 - although purity is not
U.-purity for any torsion class '1j, a torsion class is closed
under pure subgroups if and only if it is closed under 1 0-pure
subgroups, where :T o is the class of all torsion groups. Some
special cases of the general question are answered, for example
the case where each of a and tL is determined by a single rational group. We also consider an approach to the general
problem. A class of groups defined in terms of a rank function
associated with a given 1A. is described, whose members, with
those of tk, determine all ccy closed under it-pure subgroups.
When 11.= ;10'
the groups in question are the rational groups,
so the results of Chapter 3 indicate that a smaller class will
in general suffice for representations of the torsion classes 27 .
Some other examples are also given.
Chapter 5 has two brief sections. In the first we
discuss the Amitsur radical construction [1] starting from a
single rational group and in the second we characterize the
torsion classes of abelian groups whose idempotent radicals r
split, i.e. r(A) is always a direct summand of A.
Some of the results presented here have been published
elsewhere ([18], [19]). The main results of [18] are contained
in §2 of Chapter 3 while the theorem in [19] is a sort of
leit-motiv for the theory of types in the present work: it is
stated explicitly as COROLLARY 5.4, and the proof of THEOREM 5.3
is a generalization of the proof given in [19], the essential
part of which is also contained in the proof of THEOREM 4.79;
the result can also be obtained, in a rather different manner,
as a joint corollary to THEOREMS 3.12 and 3.13.
In notation and terminology we generally follow Fuchs
[15] or Mitchell [33], and most deviations have the sanction of
popular usage, but for the reader's convenience we give the
following table of notation.
agr category of abelian groups
group of integers
group of rational numbers
Q(P) group (or ring) {im,n c Z} where p is a prime
Q(S)
group (or ring) { zn C Z ,n C Z with prime
factors in S} where S is a set of primes
I(p) group (or ring) of p-adic integers
Z(n) cyclic group of order n
co Z(p ) quasicyclic p-group (p prime)
[LW group generated by set U.
[xx IA c A] group generated by set {x x IA e Al
group generated by elements
[x] * smallest pure subgroup containing x, where x is
an element of a torsion-free group
h(x) height of an element x of a torsion-free group
t(x) type of an element x of a torsion-free group
T(X) type of a rational group X
type of a height (h 1 ,h2 ,...)
G(0) subgroup {x C GIT(x) > al of a torsion-free
group G
A e B, ® A direct sum (= coproduct = discrete direct sum)
(x)
T-TAA
(aX)
[A, B]
//
direct product (= product = complete direct sum)
element of (DAA orTrAx
group of homomorphisms (morphisms) from A to B.
end of a proof
Unless the contrary is stated explicitly, "GROUP"
always means "ABELIAN GROUP".
CHAPTER 1
PRELIMINARIES
1. Torsion Theories
In this section we shall work in a locally small
abelian category a■ which is subcomplete in the following sense:
DEFINITION 1.1: A category is called subcomplete if
for every set' A1 IX EA 1 ofsubobjects of cm object A, the
direct sum (copmduct) 4)Ai and the direct product (product)
TT(AJA1) both exist.
DEFINITION 1.2: A torsion theory for a< is COI ordered
pair (a , a.), where a and 9are classes of objects of satisfying the following conditions:
(1) 27 r) = {o}. • (ii) If T ---) A —4 0 2,s exact with T c , then
A c a . (iii) If 0 A I, is exact with F c, then Ac .
(iv) For each object K of g<there is a short exact
sequence
0---)T-4K-4F-40
with T c a and F C In this situation g is called a torsion class, 2X a torsion-free
class.
We shall describe (ii) and (iii) by saying that the
relevant class is closed under homomorphic images, subobjects
respectively. Other closure properties for classes will be
described similarly. Note that both a' and are closed under
isomorphisms.
THEOREM 1.3: A non-empty class ZI is a torsion class
if and only if it is closed under homomorphic images, direct
sums and extensions. a is a torsion-free class if and only if
it is non-empty and closed under subobjects, direct products
and extensions.
(Here closure under extensions means that in an exact
sequence
0-4K-4L—)M-40
L belongs to the class if K and M do).
PROOF. [9] Theorem 2.3.//
THEOREM 1.4: Leta, 2F be classes of objects of g< .
Then (7,2F) is a torsion theory if and only if
[T,K] = 0 for aZZ T c a ‹.> K C 2P
and [K,F] = 0 for all F e 2X 4=> K ea.
PROOF. [9] Theorem 2.1.//
THEOREM 1.5: A torsion class a belongs to a unique torsion theory (3,2F), where
2F = fsi[A,B] = 0 for al/ A c 2f 1.
Similarly a torsion-free class ‘S belongs to a unique theory
), where
2.
3.
it = tAl[A,B] = o for all B c 4§
PROOF. [9] Proposition 3.3.//
From THEOREM 1.3 it is clear that the intersection of
any family of torsion (rasp. torsion-free) classes is a torsion
(resp. torsion-free) class.
DEFINITION 1.6: For any class C of objects of k, T((!)
is the smallest torsion class with C as a sub-class, F(C) the
smallest such torsion-free class. If C has a single member C,
T(C) and F(C) will be used rather than T({C}) and F({C}). T(C)
will also be referred to as the torsion class determined by C.
THEOREM 1.7: For any class C of objects of X,
= 0 for all C c C 1 is a torsion class,
{131[C,B] = 0 for all C is a torsion-free class,
T(C) = fAl[C,B] = 0 for all C c (1 => [A,B] = 01,
F(C) = IBI[A,C] = 0 for all C E C =;.> [A,B] =
Hence T(A) ci T(6) and F(A) C- F(6) whenever
PROOF. [9] Propositions 3.1 - 3.3.//
2. Torsion Classes and Radicals
DEFINITION 1.8: A fUnctor r -- akis called a
sub functor of the identity if
(i) r(K) 5i K for each object K
and (ii) for any morphism f:K-4L in a i
r(f): r(K)-4r(L) is the restriction of f to r(K),i,e. the
diagram
4 .
r(f)
r(K) -----4r(L)
K L
commutes, where the vertical arrows represent inclusions.
If in addition
(iii) r(K/r(K)) = 0,
✓ is called a radical, and an idempotent radical if also
(iv) r2 = r, is. r(r(K)) = r(K) for every object K.
Conditions (ii), (iii), (iv) are independent for functions
which assign subobjects to objects of k, as is demonstrated by
the following simple examples for a =at. .
EXAMPLE 1.9: r(A) = p A for every A c a2r, where p is
prime. Since for any homomorphism f: A-4B, we have f(pA) cip B,
✓ can be made into a functor satisfying (ii). (iii) is also
satisfied, but (iv) is not, as for example p Z(p 2) Z(p) and
p Z(p) = 0.
EXAMPLE 1.10: Define
1A if A Z r(A) =
0 if A f z . ✓ satisfies (iii) and (iv), but if 0 A Z, there are non-zero
homomorphisms f from Z to A, so f(r(Z))1; r(A).
EXAMPLE 1.11: r(A) = A[p] = Ca e Alpa = 01, where p
is prime. (ii) and (iv) are satisfied, but for example r(Z(p2))
Z(p) and r(Z(P2)/r(Z(P2 ))) r(Z(p)) = Z(p).
5 .
PROPOSITION 1.12: Let (cr,u-) be a torsion theory for 2k. Every object K has a unique largest subobject orK belonging
to a. g K satisfies the following equalities
(i)zrK .u{K , 5.; 00 cu
(ii) K = KIK/K' e .
Also, (K/aK) = 0 for every K.
PROOF. [9] Proposition 2.4.//
For any morphism f: in5C., we have f(K) IL ZTL,
since f(h() 6 J. In addition 2rano = al( for every K, so as
a consequence of PROPOSITION 1.12 we have
COROLLARY 1.13: Let (a,a) be a torsion theory forX.
For any object K let r7 (K) = grK and let rz act on morphisms by
restriction. Then r7 is an idempotent radical. Further,
r7 (K) = K if and only if K ea and r7 (K) = 0 if and only if
Conversely, each idempotent radical determines a torsion
class. Proof of this requires
LEMMA 1.14: Let r: be a radical, C the class
of objects L for which r(L) = 0. For any K e we have
r(K) = rI {K' KIK/K' e C }
PROOF. As for Proposition 2.4 of [9].//
This result seems to be well-known. It appears without
6.
proof in [5]. It is clear that r(K) = K if and only if [K,L] = 0
for each L e . Using THEOREM 1.7 we therefore obtain
COROLLARY 1.15: For any radical r: the
class
{K EXI r(K) =
is a torsion class.//
Thus in particular, if r is an idempotent radical,
ar = {K e)61 r(K) = KI
is a torsion class.
LEMMA 1.16: If r : 5(6—* X is a radical and K S.. r(L),
then r(L/K) = r(L)/K.
PROOF. [32] p.110.//
For any idempotent radical r and any K eae,
rea r(K/r(K))) 53 r(K/r(K)) = 0 .
But as r2 (K) = r(K), r(K) belongs to Zf r , so r(K) ,Z7rK and by
LEMMA 1.16,
= :,:f r (K/r(K)) = 0
i.e. r(K)
Also, for any torsion, class a ; we have the identities
cxleaK . {K c oz I r 7 (K) = K} .
We have thus proved
TBEOREM 1.17: There is a one-to-one correspondence
between torsion classes 7 • of and idempotent radicals r:2-4 2 ,
defined by
;
In [9] a stronger assertion is made, namely that a
subfunctor of the identity is an idempotent radical if and only
if its class of fixed objects is a torsion class. The discussion
above shows that this is false. A partial ordering of the
radicals 51,--) c7k is defined by
r 1 .:5r2 4=> r/ (K)g; r2 (K) for all K c 2k.
If 7 is a torsion class, its associated idempotent radical r
is (in the sense of this relation) the smallest radical whose
class of fixed objects is 21 . This is proved in [32] (p.110).
3. The Kurosh Construction
We now discuss a construction, which while valid in
more general categories, produces torsion classes in subcomplete,
locally small abelian categories.
Let O. be a category satisfying the following conditions:
(i)a has a zero object.
(ii) Every morphism of Ow has a kernel.
(iii) Every morphism has a conormal epimorphic image.
(iv) If f:A--) B is a conormal epimorphism and A'
Is a normal subobject of A, then f(A') is a normal subobject of B.
(v) Every infinite, well-ordered, strictly increasing
chain of normal subobjects of an object has a normal union.
Given a class (1). of objects of a, we define a class
by transfinite induction as follows:
7.
Let
C1 = Lk C O. IA is the image of a conormal
epimorphism from some C c(3 1 .
If Ca
has been defined for all ordinals a <8, let
C? 0 = {A e(11For every non-zero conormal
epimorphism B has a non-zero normal
subobject belonging to some ea , a < 8}.
Finally, let e =yea . 0, is called the lower radical class
determined by C. .
This construction was first used by Kurosh [27] for
rings and algebras. The possibility of generalizing it to
categories satisfying conditions (i) - (v) was demonstrated by
Shul t geifer [39]. It has since been used in the category of all
(not necessarily abelian) groups (a brief discussion appears in
[29]) and elsewhere. We have given here a slightly modified
form due to Sulinski, Anderson and Divinsky [40].
For the categories in which we are interested, the
construction takes a simpler form.
PROPOSITION 1.18: If asatisfies (i) - (v) and if in
addition normality of'subobjects is transitive ina, the lower
radical class construction terminates at the second stage, i.e. —
for every class e, e =C 2'
PROOF. We need only show that e 3 = C 2 . To this end
let A E C 3 have a conormal epimorphic image A", with B e e 2
a non-zero normal subobject of A". B has a non-zero normal
subobject B' C C3 , and by assumption, B' is normal in A". Thus 1
8 .
A c e 2 , so e 3 = e 2
PROPOSITION 1.19: Let a, be a subcomplete, locally
small abelian category. For any class C of objects of 3. T(e)
is the lower radical class which C, determines.
PROOF. If K belongs to T((?), so does any non-zero
homomorphic image K", i.e. [C,K1 0 for some C C C . This
means that K eC2'
so using PROPOSITION 1.18, we have
T(C)Si C 2 = C. Conversely, if r is the idempotent radical
associated with T(e) and if K E E! =C 2 , then [C,K/r(K)] = 0
for all C , so K/r(K) = 0, i.e. K T(C), i.e.
e =
In some non-abelian categories upper bounds have been
found for the number of steps required in the lower radical
construction: w (the first infinite ordinal) in the categories
of all groups (Shchukin [381) and associative rings [40], w 2
in the category of alternative rings [40]. In the case of
associative rings, it has been shown that w steps are sometimes
needed [23].
4. The Amitsur Construction
Amitsur [1] has discussed the following construction
in rings and in objects of certain categories:
Let n be a property of normal subobjects. Given an
object A, define
A1 = U{A' C AA' is normal and has property 11}
9.
and if Aa has been defined for all ordinals a <5, let
A = LJ A if 5 is a limit ordinal and otherwise let A be al3 a defined by the exact sequence
(*) 0 —4A81 --4 A0 --4 (A/A0-1)1--4 0 .
-
If the category is suitably chosen, there exists an ordinal A
for which AA = AA1.1. For such A, Amitsur called the subobject
AA the upper n-radical U(A,R). Extra conditions must be imposed
an n if the upper n -radical is to determine a radical in Kurosh's
sense (what we have called an idempotent radical in the abelian
case). However, if for a subcomplete locally small abelian
category we take Tr to mean membership of some class closed under
homomorphic images, then all requirements are met and we have
PROPOSITION 1.20: Let C be a class of objects in a
subcomplete, locally small abelian category, e l the class of
all homomorphic images of members of C, it the property
membership ofe 1 " and r the idempotent radical associated with
T(.(:). Then u(k,n) = r(K) for any K EY, .
PROOF. We show firstly that K t3 e T(C) for every 0.
If a = 1, this is clear from the closure properties of torsion
classes. The same remark holds for K a when 13 is a limit ordinal
and Ka e T(e) for all a <8. In the remaining case it is clear
from (*) that K c T((;) if K c T(e),, 0-1
Now let A be any ordinal with KA = K +4 . Then
KA c T(e), but (K/KA), = KA+1/KA = 0, so r(K/KA) = 0 whence it
follows that r(K) = KAM
10.
Even in abelian categories there is no finite upper
bound on the number of steps that may be needed in the Amitsur
construction. We shall consider some examples in Chapter 5.
5. Torsion Classes of Abelian Groups
For the remainder of this chapter we shall only consider
torsion theories and classes for Cljr
DEFINITION 1.21: If p is a prime, a p-divisible group
G is one for which pG = G. If G is p-divisible for all primes p
in some set P, it is said to be P-divisible. A group is called
p-reduced (reap. P-reduced) if it has no non-zero p-divisible
(reap. P-divisible) subgroups.
DEFINITION 1.22: A P -group, where P is a set of primes,
is a direct sum of p-groups, where p varies over P..
The torsion theories described below make frequent
appearances in the sequel; the notation given here will be
preserved throughout.
EXAMPLE 1.23: 610 , %). ao (resp. a o) is the
class of all torsion (resp. torsion-free) groups. The maximum
torsion subgroup of a group A will be denoted by A t .
EXAMPLE 1.24: (Z1 ,a). CI is the class of all P P
p-groups, (p is a prime). The maximum p-subgroup of A will be
11.
denoted by A.
12.
EXAMPLE 1.25: , ). V is the class of all
P P
P-groups, where P is a set of primes. The maximum P-subgroup of
A will be denoted by Ap .
EXAMPLE 1.26: (),610. 9D (resp.a) is the class of all divisible (resp. reduced) groups.
EXAMPLE 1.27: (SD ). (resp.R. ) is the class
P P
of all p-divisible (resp. p-reduced) groups, where p is a prime.
EXAMPLE 1.28:60p , cAr ). (resp.() is the class
of all P-divisible (resp. P-reduced) groups, where P is a set of
primes.
We conclude this chapter with a list of results from
[8].
PROPOSITION 1.29: For any prime p, T(Z(0) = 27 p
PROOF. [8] Lemma 2.1.//
From this it is easy to deduce
COROLLARY 1.30: J = T(A) for any non-divisible
p-group A.//
PROPOSITION 1.31: For any prime p, T(Z(yr))=
PROOF. [8] Lemma 2.2.//
These results are used to obtain a complete description
of all torsion classes 0 C V We find it convenient to
introduce a generic name for classes of this kind.
DEFINITION 1.32: A torsion class containing only
torsion groups is called a t-torsion class.
THEOREM 1.33: Let P1 and P2 be disjoint sets of primes
and Let 21 be the class of all groups of the form Ai e A2 ,
where Al is a P1-group and A2 a divisible P2-group. Then
g = TIZ(p)ip e Pi / v {z(;) I p e P 2 1)
Any t-torsion class is uniquely represented in this way.
PROOF. [8] Theorem 2.6.//
This result was obtained earlier by Kurosh [28].
Kurosh actually proved that the classes described are the only
radical classes for the full subcategory of abelian p-groups,
but THEOREM 1.33 follows easily from this. If a category CI
satisfies conditions (i) - (v) of Section 3, then so does any
full subcategory whose class of objects is closed under
conormal epimorphic images and normal subobjects, and so lower
radical classes can also be constructed in the subcategory.
Such classes need not be radical classes in CL, however. For
example, if CL is our and a is the category of countable
groups, then 6b, while a radical class in itself, is not a
torsion class in 061,-, .
A torsion class is called hereditary if it is closed
under subgroups; such a class is also called a strongly complete
Serre class.
THEOREM 1.34: The only non-trivial hereditary torsion
classes are the classes 21
13.
14.
Proofs' are given in [8], [37] and [42].//
PROPOSITION 1.35: T(Q(P)) =gip for any set P of primes.
In particular, T(Q(p)) = Jblp for any prime p and T(Q) = 00.
PROOF. The case of Q(p) is treated in [8] (Proposition
4.1). The argument is easily adapted to cover the general
situation.//
The results of this section provide examples of a
method we shall find convenient for labelling torsion classes in
the sequel.
DEFINITION 1.36: A minimal representation ofa torsion
class 21 is an equation T(e) = a, where T(C') 21 whenever
C' •
There is nothing unique about a minimal representation.
For example
T(Z(pm)) = cap = T(Z(pn)
for any prime p and positive integers m, n. We shall see further
illustrations in the next chapter.
With the complete classification of the t-torsion classes,
the following results show that the problem of classifying torsion
classes in general reduces to that for torsion classes determined
by torsion-free groups.
PROPOSITION 1.37: Let a be a torsion class, p a prime. Then either Z(p) e 21 or j .
PROOF. [8] Lemma 5.1.//
THEOREM 1.38: Let V be a torsion class. A group G belongs to ZI if and only if Gt and G/G t do.
PROOF. [8] Theorem 5.2.//
COROLLARY 1.39: Any torsion class satisfies the
equation
a = Tc( u ad U (g
15.
PROOF. [8] Corollary 5.3.//
CHAPTER 2
SOME TORSION CLASSES DETERMINED BY TORSION-FREE GROUPS
As noted in §5 of Chapter 1, the problem of classifying
torsion classes of groups has been effectively reduced to the
problem for those which are determined by torsion-free groups.
In Section 1 of this chapter, we examine the behaviour of the
simplest torsion-free groups, the rational groups, as members of
torsion classes. The torsion classes they determine are
completely described and the classification is extended to torsion
classes determined by rational and torsion groups.
Predictably, the situation with groups of rank greater
than 1 is considerably more complicated. In Section 2 we consider
some further examples of torsion classes determined by torsion-
free groups and show that these may have quite distinct
representations, for instance a group of rank > 2 can determine
the same torsion class as a rational group and non-isomorphic
indecomposable groups of equal rank > 2 may determine the same
torsion class. an contrast, rational groups determine the same
torsion class if and only if they are isomorphic).
Section 3 is principally devoted to some torsion
theories (j,a) where 27 is "large".
All torsion theories and classes in this chapter are
16.
for air
1. Rational Groups and Torsion Classes
PROPOSITION 2.1: Let (2, -ch be a torsion theory with
idempotent radical r. For every group A, r(A) is a pure subgroup.
PROOF. Let p be a prime. If Z(p) , then r(A) is
always p-divisible (PROPOSITION 1.37) and therefore p-pure. If
Z(p) e 21 , then A/r(A) has zero p-component, so again r(A) is
p-pure.//
Since rational groups have no proper pure subgroups,
we have
COROLLARY 2.2: If X is a rational group and (J,a) a torsion theory, then either X e or x ead/
PROPOSITION 2.3: Let X and Y be rational groups. Then
Y c T(X) if and only if T(X) < T(Y). Thus in particular
T(X) = T(Y) if and only if X -=" Y.
PROOF. If T(X) <T(Y), then [X,Y] 0, so by
COROLLARY 2.2, Y e T(X). Conversely, if Y c T(X), then [X,Y] 0
and every non-zero homomorphism from X to Y is a monomorphism.//
COROLLARY 2.4: IfYand xi ,ieIare rational groups,
then Y c T({Xi li e I}) if and only if T(Xi ) < T(Y) for some j C I.
PROOF. If T(Xj ) < T(Y), then Y c T(Xj ).f.72
while if Y e T({Xi li e I}), then for some j, [Xj ,Y] 0.//
COROLLARY 2.5: „q) is the smallest torsion class
containing torsion-free groups,
17.
PROOF. Let U be a torsion class containing a
torsion-free group G. Let {xi li e I} be a maximal linearly
independent set of elements of G and for some j c I let G'
be the smallest pure subgroup containing all x i with i j.
Then GIG' is rational and belongs to a. Hence Q e T(G/G , ) cl
and so A = T( Q) 5; a .//
Thus Q is the only rational group which must belong
to any torsion class containing torsion-free groups. On the
other hand, it is clear that if a torsion class contains Z, it
contains all groups.
PROPOSITION 2.6: The following conditions on a group
A are equivalent:
(i) Z C T(A).
(ii) T(A) = Wir .
(iii) A has a homomorphic image (and therefore a
direct summand) isomorphic to Z.
PROOF. (i) 4=> (ii): If Z e T(A), then T(A) contains
all free groups and their homomorphic images, i.e. all groups.
(i) <=1› (iii): If Z c T(A), then [A,Z] 4 0
and any non-zero homomorphism from A to Z has image isomorphic
to Z.//
COROLLARY 2.7: The class 21. of groups without five
direct summands is the largest non-trivial torsion class.
PROOF. A has a free direct summand if and only if
it has a free direct summand of rank 1, i.e. [A,Z] O. Thus
18.
= {Al[A,Z] = 01 which is a torsion class by THEOREM 1.7. If
A 0, it can belong to a non-trivial torsion class if and only
if T(A) is non-trivial, i.e. [A,Z] = 0.//
DEFINITION 2.8: A torsion class is called an rt.
torsion class if it is determined by a collection of rational and
torsion groups.
DEFINITION 2.9: The type set of a torsion class 21
is the set of types of rational members of 7.
It follows from COROLLARY 2.4 that a set r of types
is the type set of a torsion class if and only if it satisfies
(*) y e r, x ? y —› x e r .
DEFINITION 2.10: Let r be a set of types satisfying
(*), P a set of primes such that X is P -divisible whenever X
is rational with T(X) c r. T(r,p) is the torsion class
T(IX rational 1 t(X) E rl u fz(p)lp e PI).
THEOREM 2.11: A torsion class 21 is an r.t. torsion
class if and only if it has the form T(r,P). Such a represent-
ation is unique.
PROOF. Let 21 be an r.t. torsion class. By THEOREM
1.33 we may assume that
a = T({xi li c 1} u {Z(p)lp e P 1 } V {Z(pc°)Ip
Where P I and P2 are disjoint sets of primes and the X
i are
rational. Let r be the type set of a and P = lp e P I IX rational, X c Z7> p X = Xl,
19.
20.
We show that 21 = T(F,P).
Since T({Xi li e I}) contains all divisible groups and
all p-groups for p c P I- P (COROLLARY 2.5, PROPOSITION 1.37),
we have
{z(p )lp c P I -P} v {z(P')1p c P2 }
c il)
whence
.11gT(T({xi li c u, tz(p)lp CPl)
A straightforward application of THEOREM 1.7 yields
T({xi li c I} [Z(p)lp cP}) T(r,P).
Thus 27 = T(r,P). Now consider T(r,P) and T(E,S), where r E, say
y t E for some y e r. There is no a E E with y > a, so if X
and Y are rational, with T(Y) = y and T(X) c E, we have [X,Y] = O.
Since also [Z(p),Y] = 0 for every p, Y cannot belong to T(E,S),
so T(r,P) T(E,S). Finally, suppose T(r,P) = vr,$), with
q e P, q t S. Then [X,Z(q)] = 0 whenever T(X) e r while
[Z(p), Z(q)] = 0 for every p e S, i.e. Z(q) t vr,$), and this
contradiction completes the proof.//
It is not difficult to find torsion classes which are
not r.t. classes.
EXAMPLE 2.12. Any torsion-free homomorphic image of
I(p) is algebraically compact (e.g.[16]) so if countable must
be divisible. Thus Q is the only rational group in T(I(p)).
Since I(p) is reduced, it follows that T(I(p)) is not an r.t.
class.
This example also shows that distinct torsion classes
may have the same type set. On the other hand, distinct sets
of rational groups may determine the sane torsion class. It is
an easy consequence of COROLLARY 2.4 that if r and E are sets
of types, then
T({X rationalk(X) e 0) = 'MX rational IT(X) c El) if and only if for each y e r there is a a' c E with y > a' and for each a E E there is a y' r with a > y'. For this to
happen it is not necessary that one of F, E contain the other —
they may be disjoint; (see EXAMPLE 2.14 below).
PROPOSITION 2.13: If r, E are sets of types for which
T ({X rational I (X) c 11) = TaX rationaltr(X) c El),
and if I' has a subset r of types which are minimal i2 r ccnd
which satisfy
(**) y r y > for some
then r c_ E, r is the set of all minimal types in E and for
every a e E, > 7 for some c .
PROOF. Let 7 be any type in -17. Then 7 > a for some
a c E and a > y for some y e 1'. Minimality of y then requires
that = a = y. Thus r 5_ E. Let be the set of minimal
elements in E. If 7 e r and a E E satisfy > a. that as above.
7= a. Hence ' T. For any a e I-, there are types y e r, c IT E such that a > y > V. But then tc7 = so = r.
Finally, if a e E, then a > y for some y e r, so a > y >
for some 7 c Tad/
21.
22.
Type sets of torsion classes do not necessarily have
minimal elements:
(n) EXAMPLE 2.14: Let heights (h i(n)
, h2 , ...) be
'defined for n = 1,2,3,... by
( ) 1 if i = 2%, k = 1,2,3, ...
n _ hi
- 0 otherwise.
(n) Writing Tn for the type of (h i(n)
, h2 , ...), we have
Tn
> Tn+1
for each n. Let X a rational group of type
Tn
and 27 = T({XnIn = 1,2,3, ...}). The type set of 27 is
AJla > T n
, some nl, which has no minimal element. Note also
that
27 . T((x2n In = 1,20,...1) = T({x2n-l In = 1,2,3,...)).
and that has no minimal representation by rational groups.
2. Further Examples
We now consider some examples of torsion classes
having more than one minimal representation by torsion-free
groups, beginning with some remarks on groups A for which T(A)
is an r.t. torsion class. We first note that T(A) is an r.t.
class if and only if T(A) = T(CA) where C A is the class of
rational groups which are homomorphic images of A.
PROPOSITION 2.15: Let A be torsion-free of rank 2.
Then T(A) is an r. t. torsion class if and only if
(i) T(a) > T(X), for some non-zero a e A, X e CA'
and (ii) if p is a prime for which pA A, then pX X
for some X
23.
PROOF: If T(A) is an r.t. class, i.e. if T(A) = T(c),
then for some X e CA, [X,A] O. Let f:X--3A be non-zero; then
any non-zero a in the image of f satisfies T(a) > T(X). If
pA t A, then Z(p) belongs to T(A) = T( CA) so pX 4 X for at
least one X e
Conversely, suppose A satisfies (i) and (ii) and let
B be any one of its homomorphic images. If B A but B is
torsion-free, then B E eA . If Bt 4 0, then Bt belongs to
T(A) (THEOREM 1.38), and so does B for each prime p. If B
is divisible, it belongs to T( CA) and if not, then pA 4 A, so
pX 4 X for some X e CA, whence Bp belongs to T(X) cT( CA).
Since also [X,A] 4. 0, for some X e CA (by (1)), it now follows
from PROPOSITION 1.19 that A e T(CA), whence T(A) =
COROLLARY 2.16: Let A be a torsion-free group of rank
2. Then T(A) = T(X), where X is rational, if and only if
(i) T(X) is the least element among the types of
groups in CA
(ii) T(a) > T(X) for some non-zero a e A
and (iii) pX 4 X for every prime p for which pA A.//
These results are non-trivial: in §1 of Chapter 5
we shall construct indecomposable torsion-free groups A of
arbitrary finite rank for which T(A) is determined by a single
rational group. Furthermore such groups A can be constructed
for any rational group X which is not isomorphic to Q(S) for
any set S of primes. In the contrary case we have
PROPOSITION 2.17: A torsion-free group A satisfies
T(A) = T(Q(S)) if and only if A is S-divisible and has a direct
summand isomorphic to Q(S).
PROOF. If A satisfies the stated conditions, then
Q(S) clearly belongs to T(A), so
SEls = T(Q(s)) c: T(A) c: gDs .
Conversely, if T(A) = T(Q(S)) .25 , A must be S-divisible, with
[A,Q(S)] O. Any Z-homomorphism from A to Q(S) is a
Q(S)-homomorphism (regarding Q(S) as a ring) and so must split.//
EXAMPLE 2.12 represents a special case of the following
result:
PROPOSITION 2.18: If A is a homogeneous, indscomposable
torsion-free group lf rank > 1, then T(A) is not on r.t. class.
PROOF. Let A be homogeneous of type a. If X is a
rational homomorphic image of A, then T(X) > a and by a result of
Baer (see [15] p.163), if A is indecomposable, T(X) a. But
then [X,A] = 0.//
We have seen that two rational groups determine the
same torsion class exactly when they are isomorphic. The
corresponding statement for groups of rank 2 is false. If A
has rank 2 and T(A) = T(X), where X is rational, then
T(A) = T(X 0) X), and A may be indecomposable.
That non-isomorphic indecomposable torsion-free groups
of the same rank may determine the same torsion class is
24.
25.
illustrated by the following example, essentially due to
Jcinsson [25].
EXAMPLE 2.19: Let P,S be infinite sets of primes such
that Pr s = 0 and 5 ft Pu S, U(resp.V) the set of square-free
integers with prime factors in P(resp. in S). Let {x,y,z} be a
basis for a rational vector space and
- - 1 A = [u lxiu e U], B = [u
1 y,v1
, 5 z --(y+2)1u e u, v e v]
- - 1 C = [u
1 y, v
1 z P --(3y+z)lu e u, v e V]. 5
A homomorphism frcm B to A can be defined by y-45x, z-30,
3-(y + z)--->x, so since A is rational, we have A E T(B).
Similarly A T(C). Also A 60- B A 19C, so T(B) = T({A,B}) =
T(A g B) = T(A ED C) = T({A,C}) = T(C), but B and C are not
isomorphic.
PROPOSITION 2.20: Let A and B be torsion-free, C
a torsion group,
—) A --> B —3 C-40
an exact sequence. Then B C T(A).
PROOF: If B T(A), then C T(A). Therefore CT T(A)
P for some prime p, and thus [A,Z(p)] = O. Let x e C have order p.
There is induced a short exact sequence
0 -->B' [x] —40.
If r denotes the idempotent radical associated with T(A), then
since r([x]) = 0, we have r(B 1 ) = A. But this is impossible,
since B' is torsion-free and r(B I ) is a pure subgroup (PROPOSITION
2.1).//
26.
COROLLARY 2.21: If A and B are quasi-isomorphic
torsion-free groups, then T(A) = T(B).//
(For quasi-isomorphism see, for example, [42]).
We conclude with another example, based on a result
of Corner [7].
EXAMPLE 2.22. Let {xn
, In = 1,2,3,...) be a
basis for a rational vector space, p n , qn , tn distinct primes,
n = 1,2,3,...,
-m -1 A = [pn xn , qn (xn + xn+1)1m,n, = 1,2,3,...]
-1 B = [p-m y , t (y + y )1m,n = 1,2,3,...] nnnn n+1
-1 -1
Cn
= [p x,-m
x , pn+1 ,
y q (x +yn+1. ),t (x +y )1m=1,2,3,...],
n n n+1 n n n n n+1
n = 1,2,3,... . Clearly A and B have rank y6 and each Cn is of rank 2. A monomorphism f from A to B can be defined by
(*) 11(x 1
) = q • Y 1
tf(xn+1 ) = qn qn+1 Yn+1
In the resulting short exact sequence
0 ----> A—>13_) B i!--> 0
(*) implies that B" is a torsion group, whence by PROPOSITION
2.20, B e T(A). Similarly A E T(B), so T(A) = T(A ED B) = T(B).
But A ED B Cn2n=1,2,.. so T(A) = T({Cn In = 1,2,3,...)). It
is straightforward to show that [C m, Cn ] = 0 if m n, so both
representations are minimal.
3. Some Large Torsion Classes
An alternative method of describing a torsion class
cj is to specify a class C (preferably as simple as possible)
for which in the corresponding torsion theory (J,.) we have F(C). Thus for example consideration of classes C! of
rational groups provides further examples of torsion classes,
none of which is an r.t. class.
PROPOSITION 2.23: There is no torsion theory ( r,7,3.)
for which both r,7 and a. are determined by rational groups.
PROOF. If both 3 and are to contain rational
groups, both must be non-trivial and '0 must contain Q. Let X0
denote a rational group of type e. Suppose
a= vfx ly e r1), a= F({x la e El).
For any a a E, there is a torsion-free group G of rank > 1,
homogeneous of type a, such that every rational homomorphic
image is divisible. (See [17] p.21). For such a group G,
[G,X(19 ] = 0 for every a' e E, so G belongs to J. But for each
y a r, [X ,G] = 0, so G is in r.4. But then G = 0.//
Note that for PROPOSITION 2.23 it is sufficient to
assume that is an r.t. class.
Our next result characterizes some "large" torsion
classes in terms of cardinal numbers.
DEFINITION 2.24. A group G is called n-free, where
n > 2 is a positive integer, if every subgroup of G with rank
27.
28.
< n is free. If every countable subgroup is free, G is called
4. 1 -free.
PROPOSITION 2.25: Let
1= T({A torsion-freel[A,Z] = 0, A has rank < 11M }),
where tK = 2,3,...; )4%1 , and Let ( 4t1 , eritt ) be the corresponding
torsion theory. Then attt is the class of all tft -five groups.
PROOF. Each 0 contains p-reduced groups for every
prime p and therefore all torsion groups, so only torsion-free
groups need be considered.
(i) -th finite: Let B c 41,1‘ have a subgroup B' of
rank < 144 . Then [A,W] = 0 for each A E • If [B',Z] = 0,
then B' E Z4vt , so B' = O. If not, then B' = B1 e Z l , where
Z 1 ; Z, and B1 has rank <ltk, so as above, Bi = 0 or Bi B2 E) Z2
Z2 1" Z. Since B has finite rank, repetitions of this argument
show that B' is free and thus that B is 1n-free. Conversely,
if B is - -free, A has rank <thand there is a non-zero
homomorphism f:A B, the image of f is free, so [A,Z] O.
Hence B is in 3. (ii) itt . Xi : If B belongs to , then as in (i)
1
B is n-free for n = 2,3,..., so by Pontryagin's Theorem
(see [15] p.51), B is W 1-free. The converse is proved in a
similar way to that in (i).//
The result cannot be extended to arbitrary cardinal
numbers: let jr.ft, be defined as above,
31,1=JB1[A,B] = 0 for all A e J
29. 00
Thenmustcontainr71zeach , but this group is
i=
notti -free for ?II > )4 1 . Note also that "2-free" means
"homogeneous of type T(Z)".
All the torsion classes of PROPOSITION 2.25 are distinct.
Before proving this we note more economical representation of
them.
PROPOSITION 2.26: For 2 < til < —
27„. TM torsion-Pee 1[A,Z] = 0, A has rank
<1K and is homogeneous of type T(Z)}).
PROOF. Let r be the idempotent radical for the torsion
theory (a 2' 7.. ). Since ;1
2 c; it follows from the exact 2 41‘,
sequence
0-4 [A/r(A), [r(A), B] = 0
that for any B e ‘%, [A,B] = 0 if and only if [A/r(A), B] = 0,
and that therefore
J= T({A/r(A)1[A,Z] = 0, A is torsion-free with
rank <It })
Now if A has rank <tt, then so does A/r(A). Thus:
ih/r(A)1[A,Z] = 0, A is torsion free with
rank <111}. {A torsion-freel[A,Z] = 0, A
has rank <it and is homogeneous of type
T(Z)1S; {A torsion-freel[A,Z] = 0, A has
rank <it}
This establishes the result.//
30.
PROPOSITION 2.27: The classes aZT 2'
are all distinct.
PROOF. It is clearly sufficient to show that
gn+i for finite n.
J 2 is determined by all rational groups X with
T(X) > T(Z), so by PROPOSITION 2.26, 0 2
For each integer n > 2, there exists a torsion-free
group Gn of rank n such that
(i) Gn is homogeneous of type T(Z)
(ii) every proper pure subgroup of Gn is completely
decomposable (and therefore free)
(iii) every rational homomorphic image X of Gn has
T(X) > T(Z).
(This result is due to Corner, see [17]). Condition (ii)
implies that Gn is n-free, so Gn ZIn, but by (iii), G
n belongs
to
The functor v considered by Chase in [6] is the
idempotent radical corresponding to the torsion theory (ay, , )• .1 1
We have seen that the class V. of all groups without
free direct summands is the largest torsion class and that
F(Z)) is the corresponding torsion theory. This theory
does not have the form ( rc3 3): clearly it cannot be
(0 ) for finite n, and since there are indecomposable n n
1-free groups of uncountable rank (see [17] p.24),
3 1 .
Finally we note a connection between F(Z) and the
class 1 of torsionless groups. (For torsionless modules in
general see [24] pp. 65-69).
PROPOSITION 2128: F(Z) = F(4)
PROOF. F(Z) F(1), since Z C L. Also F(Z) contains
all subgroups of direct products of copies of Z, i.e. all
torsionless groups, so Y., CF(Z), whence F(1) F(Z).//
The class of torsionless modules over any ring is
closed under submodules and direct products so Jf) has these
properties.does not coincide with F(Z), however, because of
PROPOSITION 2.29: Lis not closed under extensions.
PROOF. If Lis closed under extensions, then in
every short exact sequence
e: 0-3 Z -7)A --)ftz.---) 0, i=1
where each Zi Z, A must be torsionless. Let f(1) = a. Then
g(a) 0 for some g 6 [A,Z]. Let g(a) = n and form the pushout
corresponding to multiplication by n in Z:
e: 0 Z -f-5, A -371 Zi-4 0
* Htz. o--.-)z-5B--)Fliz -4 0 . •
gf(1) = n so ne splits (see [31]p.72). Since Ext( i.,Zi , Z) is
not a torsion group ([34] Theorem 8), the proposition is proved.//
ne:
32.
By setting r(A) = ker f, f e [A,Z] we define a
radical r for which r(A) = 0 if and only if A is torsionless.
The last result shows that r is not idempotent. Charles [5]
w+1 w points out that Fuchs has shown that r T r , where w is the
first infinite ordinal.
33.
CHAPTER 3.
ADDITIONAL CLOSURE PROPERTIES FOR TORSION CLASSES I:
DIRECT PRODUCTS, PURE SUBGROUPS
We begin the discussion of closure properties for torsion
classes by considering closure under direct products and pure
subgroups, one section of the present chapter being devoted to
each of these two properties. There is some interrelation between
the material of the two sections; in particular, the theory of
algebraically compact groups has a central role in each.
In the first section we obtain a complete description
of the torsion classes closed under countable direct products
(i.e. direct products of countable sets of groups) and in the
second we characterize those which are closed under pure subgroups.
The latter result is then generalized to cover closure under
S-pure subgroups, where S is a set of primes.
In this chapter all torsion classes are understood to
be torsion classes of abelian groups.
1. Direct Products
The following result will be used several times in this
chapter.
LEARW 3.1: Leta- be a torsion class containing a
torsion-free group A which is not p-divisible, for some prime p.
Then I(p) e 27.
34.
PROOF. [A, I(p)] 0 ([11] p.52) so let f:A --4I(p)
be non-zero and consider B = I(p)/Im(f). B/B t as a torsion-free
proper homomorphic image of I(p) is divisible, (see [11]) and so
belongs to 21 (COROLLARY 2.5). T(I(p)) contains B and therefore
Bt (THEOREM 1.38), whence B is divisible for all primes q p.
Since in addition B belongs to T(A) C: c.7 (PROPOSITION 1.37),
a contains Bt and therefore B. Since also Im(f) belongs to
g, so does I(p).//
The principal result of this section is
THEOREM 3.2: A torsion class J. is closed under
countable direct products if and only if it is determined by
torsion-free groups.
Most of the proof of THEOREM 3.2 is contained in the
proofs of the next two results.
PROPOSITION 3.3: Let An n = 1,2,3,... be torsion-free
groups. Then
-- T({Anin = 1,2,3,.../) = T( Ift An) = T(111.11 An).
PROOF. The first equality obviously holds; since also
Am
T( - A ) for each m, we have T( 'ED An) C— a T(1-1- An).
Let f:1-TAn --4Y be a non-zero epimorphism.
If Y 0 for some prime p, then if Y is reduced, we have
pl-TAn T-TAn so pAm Am, for some m, and thus Yp e T(Am) c.=
while if Y is not reduced, then [A n , Yp ] 0 for each n.
If Y is torsion-free, then either f(Am) 0 for some
DI Or = 0, in which case f factorizes as
35.
flA >Y
T-TAni ®A
where all maps are epimorphisms. 1-TAn/n
is algebraically
compact (see [23]). Thus r-TAn/ (E)An is the direct sum of a
divisible group and a (reduced) cotorsion group [16]; so,
therefore, is Y, which being torsion-free is algebraically
compact [16]. Thus Y = D T-TR(p), where D is divisible and
R(p) is inter alia a reduced 1(p)-module. If D t 0, then
[An, Y] t 0 for each n. If D = 0, let R(p) 1 O. Then
pr-TAn fr-TAn and thus pA A for some value of in. By m
LEMMA 3.1, I(p) e T(A). Since there is an epimorphism
(actually an 1(p)-epimorphism) from a direct sum of copies of
I(p) to R(p), we have R(p) e T(A).
Thus in all cases [Am ,Y] 0 for at least one value
of m, whence by PROPOSITION 1.19, TJAn belongs to T( ED An). This completes the proof.//
PROPOSITION 3. 4 : Let a = T ( {Ax I A c A}), where each
Ax is torsion-free and let Bn , n = 1,2,3,... be torsion groups
in 27. Then 0 containsTalT B n=1 n .
PROOF. Let f:1-1Bn --3 G be a non-zero epimorphism.
If for some prime p, G is non-zero and divisible, then
[Ax,
Gp] 0 for each A c A, while if G is non-zero but not
divisible, then priBn 0 T-TBn , so pBm Bm for some m which
means that p(B ) (B ) . Since (B ) belongs to a , so do all m p m p m p
p-groups; in particular G is in LJ.
If G is torsion-free, then f( Bn) = 0, so f
factorizes as
FTBn
> G
T-TBn/ Bn
where all maps are epimorphisms. As in PROPOSITION 3.3,
G = D r-fR(p), p prime, and we need only consider the case
where D = O. If this is so, and R(p) 0, then p FIBn f Fi Bn ,
and as in the first part of the proof, Z7 contains all p-groups. Hence at least one AA is not p-divisible, so as in PROPOSITION
3.3, 1(p) belongs to awhence R(p) does also. This proves that
TTBn c
PROOF OF THEOREM 3.2. If T is determined by torsion-free groups and if fAn In = c , then (An) t and
An/(An) t C for each n. By PROPOSITION 3.3, J—TAn/(An) t
T( EDAn/(An )) cl aand by PROPOSITION 3.4, 11(An) t ea , so
from the short exact sequence
0 ---)T1(An) t —47 An--> Fir Al (An) clearly VIA e J.
Conversely, suppose a is closed under countable direct products. Clearly 0 is not a t -torsion class. If it is not
determined by torsion-free groups, then for some prime p,
36.
37.
Z(p) c 0 but all groups in 0 n 0 are p-divisible. Let
[xn ] ==" Z(pn), n = 1,2,... . Then TT [x] C a , so ,
[xn ]/(rT[xn ]) t eUn (3. Suppose p(ax) - (xn) E 0
(TT [xJ) t, an e Z. Then for some positive k e Z, pk (p(anxn)- k i_ (xn)) = 0, so pk (pan - 1) xn = 0 for all n, i.e.
pni_ (Pan
IN.
For n > k, this means that pn-k I (pan - 1), which is impossible.
Thus (xn) + ( -17[xn ]) t has zero p-height in T • Exn ]/(Tl[xn ]) t ,
contradicting the required p-divisibility of TT kn iicri- kap „II
If . {AX IX eAlc...,9 for any set P of primes, then
eA 1-1. A C o P ' without any restriction on the size of A. Whether AX any other torsion classes have this property, or the corresponding
one for ki <flj , where itt > 0 ' is not known. A related
result is
PROPOSITION 3.5 Let (!. be a class of slender groups
and
=. {GI [G,C] = 0 for all c e C }
Then 2 n is closed under direct products for which the
nunzber of components does not exceed the first cardinal number
of non-zero measure.
PROOF. Let . {GA IX cA} C r■ 30 , where A has
appropriate cardinality. Then for any C e e , [ (et Gx ,c] 0
and consequently for any homomorphism f:TTG A --) C c C ,
f( e GA) = 0. By a theorem of tog ([15]p.170), f = 0, so
TTGA c g.it
38.
In particular, e may consist of a single reduced
countable torsion-free group [36]. Additional information
concerning the structure of direct products modulo direct sums
of torsion groups, or direct products modulo their torsion
subgroups, together with PROPOSITION 3.5 should provide at least
partial answers to the questions raised above.
PROPOSITION 3.6: Let P be a set of primes andOp
the class of P -divisible algebraically compact groups. Then
T(CL2) = TUT(p)lp 4 P1).
PROOF. T(Q) 2? T({1(p)ip 4 P}), since each 1(p)
is algebraically compact. Conversely, if G belongs to OLp ,
then G = D () 1-TR(p), p 4 P, where R(p) is an 1(p)-module.
For each q 4 P, R(q) c T(I(q)), so since also D E T(I(q)),
THEOREM 3.2 implies that G belongs to T({i(p)ip 4 P}), whence
T(cLp) 5.3 T({i(p)lp 4
In view of this result and Nunke's characterization
[34] of slender groups as reduced torsion-free groups containing
no copy of any 1(p) and no direct product of infinitely many
infinite cyclic groups, there seem to be good grounds for the
following
CONJECTURE: (T(CL), F(c)) is a torsion theory, where
ais the class of all algebraically compact groups, J the class
of all slender groups.
If (1.(1),3) denotes the torsion theory for T(Q), then by PROPOSITION 3.6, a consists of those reduced torsion-free
groups containing no copy of any I(p). Note also that
F(g), since while Qe is closed under extensions and
subgroups, it is not closed for products — Z is slender but
direct products of copies of Z are not.
2. Pure Subgroups
As a first step in the discussion of closure under pure
subgroups, we show that every torsion class with this property
is either a t-torsion class or an r.t. torsion class.
PROPOSITION 3.7: All t-torsion classes are closed
under pure subgroups.
PROOF. Let S 1 , S2 be disjoint sets of primes. If A /
is an S 1-group and A2 a divisible S 2-group, then clearly any
pure subgroup of A l e A2 is the direct sum of an S 1-group and
a divisible S2-group.//
THEOREM 3.8: A torsion class a is closed under pure
subgroups if and only if 27 n ao is.
PROOF. Let A' be a pure subgroup of A c a, and consider the induced diagram
0 0
0 ---4 A' —4 A' ---4 AVA'---> 0
g
0 At A --)A/At
0
39.
40.
with exact rows and columns, where g is defined by g(a ?+A;) = a'+At .
A't is pure in A' and hence in A. Therefore tq is pure in A t , so
by PROPOSITION 3.7, iq E 27n 'J o .
The kernel of g is A' (1 A t/A = O. If, for some
non-zero n c Z, a ° c A' and a e A we have g(a'+A) =
then m(a'-na) = 0 for some non-zero m e Z, i.e. ma' = mna. Since
A' is pure in A, there exists a" E A' with ma' = mna". But then
g(a'+A) = ng(a"+A;), so that g is a pure monomorphism. Thus if
a ngb is closed under pure subgroups, A'/A; e n :; 03 , so
A' c 27 and 0 is therefore closed under pure subgroups.
The converse is obvious.//
THEOREM 3.9: If a torsion class a is closed ?eider
pure subgroups, then
.J= T(( 0 n '3 ( ) u a )
where 27 is the class of rational groups in T.
The proof uses the following lemmas:
LEMMA 3.10: For (c."1 and a as in THEOREM 3.93
".40 = ) n
PROOF. Clearly T(:1) ,) c: o n r:;1, 0. Let Abe any
group in On r500 . Then A is a homomorphic image of (E) [a] *
where the direct sum extends over all a E A. Thus A c T(g),
since each [a] * E ad/
LEPRA 3.11: For any two classes C I , e 2 of gryups, ve 1 4.0 e2 ) = Ter(e 1) T(
We omit the proof of this result, which consists of a
simple application of THEOREM 1.7.
To complete the proof of THEOREM 3.9, we observe that
a= T[( g 0 ) ( n a'o)] = 11( f■ rjo )k-,
(T(Z1) n `.0)] T [ ( 3 n 'J o ) u T(J)] = ao )
Let XI , Xn be rational groups with types
respectively. Then any (xl , E X1 (i) ® Xn , with
x xn
0 has type x1
r0:11 . 1 , Thus one requirement if
an r.t. torsion class T(F,P) is to be closed under pure subgroups
is that r satisfy
(*)
so if X and Y are rational groups with incomparable types,
T({X,Y}) is not closed under pure subgroups.
Now if T(r,P) is an r.t. torsion class for which r
satisfies (*), then for every torsion-free group G,
G(r) = fx E GIT(x) E
is a pure subgroup, since if x, y c o(r) we have
T(x-y) > T(x) T(y) c r co is regarded as having a type greater
than all others) and T(nx) = T(x) for n c Z. o(r) is also
functorial in an obvious way. The following theorem describes
a connection between the functor ( )(r) and the idempotent
radical associated with T(r,p).
THEOREM 3.12: Let T(r,P) be an r.t. torsion class
for which r satisfies (*), r its idempotent radical and
C(r) = fo torsion -free1G(r) =
41.
42.
Then the following conditions are equivalent:
( i) C (r) is closed under extensions
(ii) G(F) = r(G) for all torsion-free groups G
(iii) T(T,P) is closed under pure subgroups.
PROOF. (i) => (ii): Let G be torsion-free and let
G' be the subgroup of G defined by the short exact sequence
o —3c(r) --) (G/G(r)) (r) —4 O.
If Cl(r) is closed under extensions, then G' e C(r) and since
for x e G' we have T G ,(x) <TG (x), it follows that G'c o(r),
whence (G/G(r))(r) = 0, and [X,G/G(r)] = 0 whenever X is
rational with T(X) c F. Also G/G(r) is torsion-free, so
r(G/G(r)) = O. Since o(r) E T(r,p), therefore, o(r) must be
r(G).
(ii)=> (iii): If cm = r(G) whenever G is
torsion-free, then T(r,P) 0 = C (r), which is closed under
pure subgroups. By THEOREM 3.8 T(r,P) is also.
(iii)==> (i): Let T(r,P) be closed under pure subgroups
with G e T(r,P) torsion-free. Then [x] * e T(r,P) so t(x) =
t([x] *) c r for all x e G, whence G e C (r) , i.e. T(r,P) n 73 0
C (r). Since the reverse inclusion also holds, C(F) =
T(r,P) ao is closed under extensions.//
THEOREM 3.13: A torsion class a is closed under pure
subgroups if and only if either
(1) a is a t-torsion class
or (ii) g = TaZ(p)lp e P} u {Q(S)}) where P and S
are sets of primes with P( S.
For the proof we need
LEMMA 3.14: Let {XxIX E A} be a set of rational groups
and S = fp priffelpxx = Xx for each A e Al. If T({Xx IX e A}) is
closed under pure subgroups, then it contains Q(S).
PROOF OF LEMMA. Let p l , p2 , p 3 , ... be the natural
enumeration of the primes, let J = fjlp i Sl and denote Xx
by A. For each j 6 J, choose a C A with h.(a ) = 0, where h. J J
denotesheightatpr Forexample,lota..(xix) with xix E Xx
satisfying the following conditions: (i) 0 for some p E A JP
f J JP JA
for A p.. For a natural number i J, let ai be an arbitrary
element of A, and regard the resulting (a i ) as an element of
= 1,2,3,..., where each A = A. h((a.)) = h(ai ), i=1
the former height being taken in T-FAi' the latter in A. In
P J
S-divisible, the height of (ad at a prime p is infinite if
p C S and zero otherwise, i.e. T((ai)) = T(Q(S)) and 1-TA has
a pure subgroup isomorphic to Q(S). By THEOREM 3.2,
T-TA E T(IXAIA CAI), which if closed under pure subgroups
must therefore contain Q(S).//
Since each Xx is S-divisible and T(Q(S)) is the class
of all S-divisible groups (PROPOSITION 1.35) we have
COROLLARY 3. 15 : With the notation of LEMMA 3.14, if
T({Xx IA e A}) is closed under pure subgroups, it is the class of
all S-divisible groups.//
43.
44.
PROOF OF THEOREM 3.13. Let a be a torsion class closed under pure subgroups. If IJ is not a t-torsion class,
let r be its type set and for each y e F let X be a rational
group of type y. Then
U. Tc(gn 21 0) v {Xy r), c r})
(THEOREM 3.9)
and
gnao T({Xy c r}) rl 3- 0 (LEMMA 3.10)
By THEOREM 3.8, T({Xy ly E r}) is closed under pure
subgroups and therefore, by COROLLARY 3.15, is the class of all
S-divisible groups, where S is the set of primes dividing (E)Xy .
Thus
27 = T(( Jr go) {Q(0}).
Let P = {p C SIZ(p) E g }. Since T(Q(S))c2 a, a 00
contains the groups Z(p ) for all primes p as well as Z(p) for
primes p 4 S. Thus by THEOREM 1.33 and LEMMA 3.11,
T({Z(p)lp S} {Z(p)lp c P} u {Z(pc°)I all p}u{Q(S)})
= T({Z(p)ip e Old {vs)}).
Conversely, that any class
0 = T({Z(p)Ip e u {Q(S)})
with PC: S is closed under pure subgroups follows from THEOREM 3.8,
LEMMA 3.10 and the observation that T(Q(S)) is closed under pure
subgroups. By PROPOSITION 3.7, the proof is now completed/
Note that by THEOREM 1.7, for a torsion class which
is not a t-torsion class, the representation
= T({Z(P)IP c P} V {Q(S)})
Is unique. Our next result characterizes the groups in such a class.
45.
PROPOSITION 3.16: A group G belongs to
27 . T({Z(p)ip E P} L {Q(S)})
where P and S are sets of primes with PS-7... S, if and only if there
is a short exact sequence
G ° --4 G"-40
where G° is a P-group and G" is S-divisible.
PROOF. Let G c 27 and G ° = 89G where the direct sum
extends over all p e P, G" = GIG'. Then W; has no P-component
and belongs to 2 (THEOREM 1.38) so therefore has divisible
S-component. Thus G'; is S-divisible. G"/W; is torsion-free and
belongs to T. If not S-divisible, it has a non-zero S-reduced
torsion-free homomorphic image B. But then B E 27 anI [Q(S),B] =
0 = [Z(p),B] for each p C P and this contradicts THEOREM 1.7, so
G"/G' is S-divisible, whence G" is also. The converse is obvious.//
DEFINITION 3.17: A subgroup G° of a group G is S-pure,
where S is a set of primes, if G'n nG = nG ° for every n in S*,
the multiplicative semigroup generated by S. If S has a single
element p, s-purity is called p-purity.
THEOREM 3.13 can be generalized fairly easily to describe
the torsion classes which are closed under S-pure subgroups. We
shall need
LEMMA 3.18: Let P be a set of primes. Then T(Q(P)) =
is closed under S-pure subgroups if and only if P CI S.
PROOF. If P 4 s, then Q(P s) is an S-pure subgroup
of Q(P), but Q(P c S) 4: A, • The converse follows from the fact
that S-purity implies P-purity if P C S.//
THEOREM 3.19: A torsion class la is closed under
S-pure subgroups if and only if either
(i) g is a t-torsion class such that j n a is
hereditary for p S
or (ii) = T({Q(P)} U {Z(p)ip E R}) , where RC._ P C. S.
PROOF. Since pure subgroups are S-pure, only the
classes described in THEOREM 3.13 need be considered.
If 7 is a t-torsion class, then clearly 21 is closed
under S-pure subgroups if and only if j n ap has this property
for every prime p. Now 0 a is either {0}, a or a (1L-T so interest is centred on g.) n 7. If p c S, then in oco n S-purity is equivalent to purity whence oaci Up is closed. If
p S, any exact sequence
0 --) Z (p)--) Z (pc) --) Z (pe°)----) 0
is S-pure, so )11)n -T is not closed. p
If 23 = TaQ(P)} u {Z(p)lp c R}) where RC, p, then as
in the proof of THEOREM 3.13, n 3'0 = T(Q(P)) r):1- 0 , so by
LEMMA 3.18 we may assume PC S. If A' is S-pure in A ;I , then A l is S-pure in A which, as a direct summand of A
t' is in
a na by THEOREM 1.38. Thus A E rs.7 S Since Z(p) E
T(Q(P)) for every p S, we have 'a s = T(Q(P))r,
Analogously with THEOREM 3.8, it can now be shown that A l /A;
has a natural S-pure embedding in A/As and so belongs to g n Thus A' E
46.
> 47.
CHAPTER 4
ADDITIONAL CLOSURE PROPERTIES FOR TORSION CLASSES II:
GENERALIZED PURE SUBGROUPS
In this chapter we consider torsion classes closed under
generalized pure subobjects in the sense of [41]. Specifically,
we consider 'U-purity where 11. itself is a torsion class. The first section is a summary of the relevant results from [41]. In
the second we obtain some results concerning the idempotence of
products and intersections of idempotent radicals. The question
of idempotence of products has some relevance to the material of
Section 3 in which a generalization of THEOREM 3.8 is obtained
for certain abelian categories with global dimension 1, purity
being replaced byli -purity.
The remainder of the chapter deals with torsion classes
for CIS& only, though many results have obvious generalizations to
module categories.
A prerequisite for a generalization of THEOREM 3.9 is
a class of groups to take on the role played by the rational
groups in Chapter 3, i.e. given a torsion theory (ti,S), we
need a class of groups in 'S whose members, together with those
of 11, determine all torsion classes closed undertIL-pure
subgroups. Such a class of groups is introduced in Section 4;
the groups are described in terms of a rank function associated
with (11,`S) which coincides with the standard (torsion-free)
rank in the case of ( ). The groups we consider are those 0' 0
of generalized rank (It-rank) 1.
In Section 5 we investigate the structure of groups
oftl -rank 1. Because of the purpose for which these groups
have been chosen, some emphasis is placed on the kinds of
1L-pure subgroups they can possess. It is shown that they
cannot be mixed and cannot be direct suns of infinitely many
subgroups. They can however be infinite direct products.
In Section 6 we consider some examples. The groups
of c0 -rank 1 (p prime) are completely described and this leads
to a representation of the groups of gs-rank 1, where S is a
set of primes. The groups of 1L-rank 1 are also characterized
when'll is determined by divisible torsion groups. Some
additional similarities between rational groups and groups of
..„SO -rank 1 are also noted.
In the final section we solve the following special
case of our general problem: to find conditions on rational
groups X and Y, necessary and sufficient for the closure of
T(X) under T(Y)-pure subgroups.
1. Generalized Purity
Of the several well-known characterizations of pure
subgroups, the one given by the next proposition has the
advantage of being element-free and of therefore suggesting
generalizations of the notion of purity to categories.
PROPOSITION 4.1: A subgroup A of a group B is pure
if and only if for every groupGwithACGCBand C/A finite,
48.
49.
A is a direct summand of G.
PROOF. [15] p.82.//
PROPOSITION 4.1 may be paraphrased as follows: a
short exact sequence
(*)
0 -->A--->B —4C —40
of groups is pure if and only if for every finite subgroup C'
of C the pullback induced by the inclusion C'--4 C gives a
commutative diagram
0 --->A--->A
* * ) 0 ---9A---> B
in which the top row represents the natural splitting. The
obvious generalization is to substitute some class e of groups
(or of objects in a suitable category) for the class of finite
groups and define a short exact sequence (*) to bee -pure if
every subobject C' of C which belongs to e gives a diagram (**).
In this section we shall discuss the theory of
generalized purity due to Walker [41] in the setting of a
subcomplete locally small abelian category A which has enough
projectives and for which Ext n (A,B) is a set for all objects
A,B E
DEFINITION 4.2: Let (.), be a class of objects of 3(
closed under homomorphic images. A subobject A of an object B
in 'X is said to be C. -pure if A is a direct summand of every subobject V of B with Ac; B' and WhIl e C. A short exact
sequence
50.
0-4A-4B--4C —40
in A, is C -pure if the image of f is C -pure in B.
THEOREM 4.3: The class of e -pure short exact
sequences is a proper class.
PROOF. [41] Theorem 2.1.//
For a discussion of proper classes see [31] pp. 367 ff.
DEFINITION 4.4: For objects A,C of k, Pexte (C ,A) is
the group of equivalence classes of C -pure short exact sequences
0--)A —>13--)C—)0
DEFINITION 4.5: An object K of 3( is C -pure
projective if for every C -pure short exact sequence
0-4A---)13--)C -4 0
the induced sequence
0 -4 [K,A]--)[K,B]--> [K,C] —4 0
is exact.
THEOREM 4.6: Let 63 be the class of projective objects
of 3L. Then K is C -pure projective if and only if it is a
direct summand of a direct sum of members of Pu C.
PROOF. [41] Theorem 2.5.//
If has global dimension 1 (and thus in particular
if
=a9.-), thee -pure projectives have an alternative description:
THEOREM 4.7: If a has global dimension 1, on object
K is e -pure projective if mid only if K = L e 14, where L is
projective and M is a direct summand of a direct sum of objects
in C.
PROOF. [41] Theorem 2.6.//
THEOREM 4.8: A short exact sequence
0—> A-4 B C
ise -pure if and only if the induced sequence
O-4 [K,A] [K,C]---> 0
is exact for every C. -pure projective K.
PROOF. [41] Theorem 2.7.//
As examples of generalized purity in CUL- , we have
S-purity for a set S of primes, in which case C is the class
of finite S-groups, and for an infinite cardinal numberfn, the
In-purity introduced by Fuchs [14] where is the class of
groups G with IGI <Th.
The generalized notion of purity has a dual:
DEFINITION 4.9: Let 6.1) be a class of objects of 'a
which is closed under subobjects. A subobject A of B is said
to be61-copure if A/A' is a direct summand of B/A ° whenever
C A and AlAy Efi. A short exact sequence
is called B-copure if the image of f is 65-cop:Axe in B.
The results given above fore -purity can be dualized,
with -copurity replacinge -purity. In particular, since the
postulates for a proper class are self-dual, we have
51.
THEOREM 4.10: The class of B-copure short exact
sequences is a proper class.//
DEFINITION 4.11: For A,C in a, copext n, (C, A) is the
group of equivalence classes of -copure short exact sequences
0-3 A-4B---> C-4 0.
In the sequel we shall be principally concerned with
the case where e is a torsion class. The pure and copure short
exact sequences associated with a torsion theory are described
by
THEOREM 4.12: Let (j 5.) be a torsion theory for 31,
and let r be the associated idempotent radical. The short exact
sequence
0 —4A--->B-4C-90
is 2T -pure (resp. a -copure) if and only if the induced sequence
0 ---) r(A) r(B)-4 r(C)---4, 0
(resp.
0 --4A/r(A)---> B/r(B) C/r(C) --) 0)
is splitting exact. As a consequence, we have
(i) r(A) = A n r(B)
and (ii) r(B/A) = (r(B)+A)/A
whenever A is either J -pure or 4'.:A -copure in B.
PROOF. [41] Theorems 3.4, 3.9 and Corollaries.//
DEFINITION 4.13: For a torsion theory (3 ;3), a short
exact sequence which is both j -pure and g--copure is said to be
52.
53.
(g,a) -biloure and for any A,C, Pext (C,A) Copext (C,A) is 7
denoted Bipext (g ) (CA).
Since the intersection of two proper classes is a
proper class, it is clear that the (:7,I)-bipure short exact
sequences make up a proper class.
2. Idempotence of Composite Radicals
As noted in §5 of Chapter 1, a group A belongs to a
torsion class 0 if and only if both At and A/At do. This
result suggests the problem of determining whether there are
any other torsion classes besides 270 for which the corresponding
statement is true for all 27 or for some given 7. For this
section we shall work in a subcomplete locally small abelian
category a . a l and 272 will denote torsion classes ina , r 1
and r2 their associated idempotent radicals.
DEFINITION 4.14: Let u and v be subftinctors of the
identity. The subAnctor u n v is defined by
(u v) (K) = u(K) ,- v(K)
with action on morphisms being determined by restriction.
PROPOSITION 4.15: If u and v are radicals, then so
are uv and u n v.
PROOF. [32] p.110.//
Thus in particular r 1 r2 is a radical, but as we shall
see, not necessarily idempotent.
54.
PROPOSITION 4.16: The statement
(*) K C 272 4=> r I (K), K/ri (K) e a2
holds for every K E ;Rif and only if r 1 r2 is idempotent.
PROOF. If (*) holds, then for every K e , r2 (K)
belongs to 27 2 so r1 r2 (K) does also, i.e. r 2 r 1 r2 (K) = r 1 r2 (K),
or since K is arbitrary, r 2 r 1 r2 = r 1 r2 , so that
r 1 (r2 r1 r2) = r 1 (r 1 r2 ) = r1 r2 ,
i.e. (r 1 r2 ) 2 = r 1 r2 . Conversely, let (r 1 r2 )2 = r 1 r2 . Then
for any K
r 1 r2 (K) = r 1 r2 r 1 r2 (K) r2 r 1 r2 (K) r2 (K)
i.e. r 1 r2 = r2 r 1 r2 . Thus if K E 27 2 , we have
r 1 (K) = r 1 r2 (K) = r2 r1 r2 (K)
which is also in a 2 . Since;r2 is closed under homomorphic
images and extensions, the proof is complete.//
COROLLARY 4.17: If r 1 r2 = r2 r 1 then
K e 2 4=> r l (K), K/r 1 (K) C7 2
and
K C 4=> r2 (K), K/r2 (K) e
PROOF. If r l r2 = r2 r l' then
(r 1r2 ) 2 = r 1 (r2 r 1)r2 = r 1 (r 1 r2)r2 = (r 1 r 1)(r2 r2) = r 1 r2
and similarly (r2r 1 ) 2 = r2 r 1 .//
The problem is therefore, in part, that of finding
commuting pairs of idempotent radicals, and in particular of
finding those idempotent radicals which commute with all others.
The next two propositions give some examples forX
55.
PROPOSITION 4.18: Let X =a- and let a l be
hereditary. Then for any 21 2 , r 1 r2 = r2 r 1 = r 1 n r2 .
PROOF. For any group A, we have
(i) r1 r2 (A) = r 1 (A) in r2 (A)
and (ii) r2 r 1 (A) g r 1 (A) ctr2 (A)
so r2 r 1 (A) = r; r 1 (A) r2r 1r2 (A). Also, r 1r2 (A)c; r I (A), whence
r2 r 1 r2 (A) ci r2 r 1 (A). Thus r2r 1r2 = r2r 1 .
Now r 1 is either the identity functor, in which case
there is nothing to prove, or r 1 assigns to each group A the
subgroup el) S A ' where S is a fixed set of primes. Since pc p
(r2A) t a2 (THEOREM 1.38), its direct summand r 1r2(A) is also.
Thus r2 r 1 r2 (A) = r 1r2 (A), so
r2 r 1 = r2 r 1 r2 = r 1 r2 = r 1 n r2 .1I
aI need not be hereditary, however:
PROPOSITION 4.19: Leta =Glir =.Z. Then for any
a2 , r 1r2 = r2 1- 1 .
PROOF. Case (1): 27 2 contains torsion-free groups.
In this case a l C:2. 2 , so for any group A, we have r2r 1 (A) =
i.e. r2 r 1 = r l . Also r 1 (A) E r2 (A) and r 1r2 (A) ç r I (A), whence
r 1 (A) = r(A) C r1 r2 (A) C r 1 (A) 1 — —
so that r 1r2 = r 1 = r2r 1 .
Case (2): 27 2 is a t-torsion class. r 2 (A) has
the form pes rp (A), where S is a fixed set of primes and for each
p c S rp(A) is either A
p or its divisible part. Thus r 1r2 (A) =
pcSr rp(A). If r(A) = Ap
, then r 1 rp(A) = rpr i (A) as in
PROPOSITION 4.18, and if not then r irp (A) = rpr i (A) = r(A) as
in Case (1). Thus
r1r2(A) = peSr rp (A) = (I) peS
r r1 (A) = r2r1 (A) p
i.e. r 1r2 = r2 r 1 .//
Let r be the idempotent radical associated with the
torsion class 27
PROPOSITION 4.20: r 1r2 is idempotent if and only if
r ir2 r.
PROOF. Let r 1r2 be idempotent with torsion cl4asst.
Then for every K C tl ra2 , we have r 1r2 (K) = r 1 (K) = K, i.e.
K ell., so a n g2 Q11. Since 1
r 1 r2 (L) = r 1r2 r 1 r2 (L) r2r 1r2 (L)c; r 1r2 (L)
for any L, we have r2 r 1r2 = r 1 r2 , so in particular r2 (L) = L
if L cli,, i.e. 1.1. c 21 2. Since also for every L C /L,
2 r 1 (L) = r 1 (r I r2 (L)) = r ir2 (L) = r 1 r2 (L) = L,
we have 11,5ig 1' so 11,= a 1 (-1 a 2 and r 1 r2 = "
The converse is obvious.//
Using COROLLARY 4.17, we obtain
COROLLARY 4.21: r 1r2 = r2r 1 if and only if r 1 r2 and
r2 r 1 are both idempotent, in which case r 1r2 = r2 r 1 = r.//
56.
PROPOSITION 4.22: r l n r2
is idempotent if and only
PROOF. If r1
r2
is idempotent, then its torsion
57.
class is cj, n 72 since r 1 (K) A r2 (K) = K if and only if r (K) =
K = r2 (K). The converse is obvious.//
Although an object is left fixed by r 1 r2 exactly
when it belongs to a l rI rj2 , this does not mean that r i m r2
must be idempotent (cf. §2 of Chapter 1).
LEMMA 4.23: If r 1 r2 = r 1 ,1 r2 , then r2r 1 is idempotent.
PROOF. r2 r 1 r2 r1 = r2 ((r 1 r2 )(1. 1 )) = r2 ((r 1r 1 ) (r2r 1 ))
= r2 (r 1 (r2r 1 )) = r2 (r2 r 1 ) = r2 r 1 . "
PROPOSITION 4.24: If any two of r 1r2 , r2r 1 , r2 r1
are idempotent, then r 1 r2 = r2 r 1 .
PROOF. If r1r2
and r2r1 are idempotent, then
r1 r2 = r = r2 r 1 (COROLLARY 4.21), while if r 1 r2 and r i n r2 are
idempotent, then by PROPOSITIONS 4.20 and 4.22, r 1r2 = r 1 n r2 ,
so by LEMMA 4.23, r2 r 1 is idempotent.//
We now give an example to show that r 1r2 and r2 r 1 need
not be equal. Note that by COROLLARY 4.21 this is sufficient to
show that idempotence is not preserved by products in general.
EXAMPLE 4.25: We consider a group which has been
discussed by ErdOS [12] and de Groot [20], [21]. Let ix,y1 be
a basis for a 2-dimensional rational vector space, and let
G = q-ny, t-n (x+ys )In = 1,2,3,...
where p, q and t are distinct primes. Let r 1 and r2 be the
idempotent radicals for and T(G) respectively, From ap
58.
examination of the type set of G (see [20] p.295), it is clear
that r 1 (C) = [p-nxin = 1,2,3,... ] Q(p). Let f be any
homomorphism from G to r l (G). Since r 1 (G) has no non-zero
elements of infinite q-height or t-height, we have f(y) = 0 =
f(x+y) = f(x)+f(y). But then f(x) = 0 and thus f = O. Hence
[G,r1 (G)] = 0, so r2r1 (G) = O. But r 1r2 (G) = r1 (G) Q(p).
This example also shows that idempotence need not be
preserved by intersections. Let r be the idempotent radical
associated with glpn T(G). Then r
1 (G) =7, Q(p) iccO n T(G)
so r r1 (G) = O. Since r(G) C , a we have r(G) gr
1 (G), so P
r(G) = r2 (G) C r r 1 (G) = O. Therefore r(G) = 0, while
r1 (G)n r2(G) = r
1 (G) Q(p). By PROPOSITION 4.22, idempotence
of r 1 n r2 would require r1 n r2 = r.
To conclude this part of the discussion we give a
"local" construction, using transfinite induction, of the radical
associated with the intersection of a set of torsion classes.
This construction was used by Leavitt [30] for radical classes
of associative rings. In an abelian category the fact that all
subobjects are normal allows a minor simplification of the
original argument.
Let {J 1 i E be a set of torsion classes inX.,
{r the associated set of idempotent radicals and K ca.
Define L 1 = K and for an ordinal number 0, assuming La has been
defined for all a <0, define
a La if 8 is a limit ordinal. Otherwise
(**) Lo =
r0-1
) for some i e I (if such exists)
such that r1(L81) 4 113_1.
Since the La's form a set, there exists an ordinal number y such
that r1 (L1) = L for each i c I.
THEOREM 4.26: For any K C 3Z., let La be defined by
(**). Then there exists an ordinal number y such that r(K) = Ly ,
where r is the idempotent radical for
PROOF. As noted above, there exists y such that for
each i E I, r1 (L) = Ly , i.e. Ly e ai . Thus r(L) = Ly , so
L C r(K). We show, by transfinite induction, that r(K)C: L.
Trivially r(K) C L l , so assume r(K) C La for each a < a. If a is a limit ordinal it is clear from (**) that r(K) C L o. If a
is a successor, then by assumption r(K) C L0_ 1 , while since
r(K) e a i we have r(K) C r1 (L13-1) = La . Thus r(K) C L for a every a, so r(K)c Ly .//
It is not clear whether the construction can always be
achieved in a finite number of steps in ac even when the number of torsion classes involved is finite. With r
1 and r
2 as in
EXAMPLE 4.25, at least three steps are needed: for G we have
G r 1 (G) r2 r 1 (G) = 0 = r(G).
3. A Simplification of the Problem
In this section we shall work in a subcomplete abelian
category Xsatisfying the same conditions as in Section 1 and
in addition having global dimension 1.
PROPOSITION 4127: Let (7,3.) and (1C,S) be torsion
theories fora, with associated idempotent radicals r, s
59.
60.
respectively such that sr is idempotent. Then g is closed under (iL ,4!1)-bipure subobjects.
PROOF. Let K' be (tL, k,5)-bipure in K e J. Then
there are split exact sequences
0 -4-8(K') ---) s(K) s(K/K') —4 0
and 0 --)10/s(10)--) K/s(K) --)(K/K')/s(K/10)-4 0.
By PROPOSITION 4.16, both s(K) and K/s(K) belong to a and so their direct summands s(K') and K'/s(10) do also. Hence K' e j .//
Note that this proposition is true without any
restriction on the global dimension of gL.
For 'U. -pure subobjects there is a result analogous to
THEOREM 3.8. Proof of this requires
LEMMA 4.28: A short exact sequence
(*) A —1 B C 0
isILL-pure, for a torsion class IL, if and only if'the induced
sequence
(**) 0 —4[K,A]--->[K,B] (K,C1-----> 0
is exact for every K
PROOF. By assumptions on:X, THEOREM 4.7 and the
properties of torsion classes, an object L isi4 -pure projective
exactly when it has the form M Q) K where M is projective and
K EU-. The sequence (*) induces a morphism.
[L,B] [11,13]QP [K,13] LT.-10[11,c] 4E0 [K,c] [L,C].
Since M is projective, f is an epimorphism, so if (**) is
assumed exact for every K c1.L , f g is an epimorphisn, so
61.
by THEOREM 4.8, (29 is #11. -pure.
The converse is obvious.//
COROLLARY 4.29: Let (713) be a torsion theory for
a., K' C K . Then K' isU-pure if ond only if KIK' .
PROOF. For U Eli the induced sequence 0 = [ U,K]-- [ u,K/Kt] 0
is exact if and only if [ U,K/K1 = O. By LEMMA 4.28, K' is
11-pure in K if and only if [ U,K/10] = 0 for all U c ILL.//
In a similar way we can prove
COROLLARY 4.30: Let s be the idempotent radical for
the torsion class U. Then s(K) is U -pure in K for all K c .1/
PROPOSITION 4.31: Let (7,,3) and (11,5) be torsion
theories fora, with associated idempotent radicals r co2d s.
If 0. is closed underil-pure subobjects, then sr is idempotent.
PROOF. Since for any K e`74, s(K) is all -pure
subobject, in particular sr(K) is alwaysli. -pure in r(K). By
assumption onU , therefore, we hf.ve rsr(K) = sr(K) for each K.
But then
(sr) 2 = s(rsr) = s(sr) = s 2 r =
THEOREM 4.32: Let (J ,3) and ('ti,S) be torsion
theories for R. with associated idempotent radicals r,s respectively.
Then 2 is closed underli. -pure subobjects if and only if sr is
iclempotent and 2 (-1 16 is closed under1,1-pure subobjects.
PROOF. We first show that zr (1 11 is always closed
underll-pure subobjects. If K' isli-pure in K cJ AU , then
K/K' clLand the sequence
0 K'-9 K 0
is split, by definition of iUL -purity. Hence 10 c a ()ll. Now suppose a n S is closed underlk -pure subobjects
and sr is idempotent. If M' -pure in M C a we have a
commutative diagram
o s (m')--4 m' —4 mi /s (m 1 )--4 o
___45 (m) m --4 m/s(m) —40
with exact rows.
f is a monomorphism, having kernel M I /s(M I ) A N, where
N is the kernel of the natural map from M/s(14') to M/s(M), i.e.
N = s (M) /s (1,1 1 ) and thus
(M I /s(1,0)) n N = (M' s(M))/s(le) = 0,
by THEOREM 4.12.
THEOREM 4.12 also says that (s(M) + M')/10 =
SO
s (M)+M' (WM' ) / ( (s (M)-I-M' ) /M‘ )= (M/M') /s (M/M' )
Hence the sequence
—) (s (14)+14') is (M) Mis (M) —) M/ (s (M)-114 1 )--4
is.-pure exact, whence as M/s (M) e geI5 , it follows that
(s (M)-1-1•1 1 ) /s (M) e a n\b • But as
11/40/s(14')=MV(M' r s(M))Z(M'+s(M))/s(M),
this means that M'/s(4') c
Also, s (M) E iJ ,i U. (PROPOSITION 4.16) . Since s (M' )
62.
63.
isik-pure in M' and the /It-pure short exact sequences form a
proper class (THEOREM 4.3), s(M t ) is/A.-pure in M and hence in s(1),
so s(M t ) e Un 'tj, . to is therefore in 7, as both s(M 1 ) and
M T /s(M t ) are, i.e. is closed underl,L-pure subobjects.
By PROPOSITION 4.31 the converse is obvious.//
CCIMI17.;:1 4.29 shows that in purity and J o-purity
coincide, so as a consequence of THEOREMS 3.8 and 4.32, we see
that in Ws, a torsion class is closed under go-pure subgroups
exactly when it is closed under pure subgroups, which raises the
question: if C is homomorphically closed, when is closure of
a torsion class under C -pure subobjects equivalent to that for
g-pure subobjects for a torsion class a ? This question is
related to the problem of determining projective closures, for
e and 0 satisfy the condition in particular when C -purity
coincides with 21 -purity, i.e. C and a have the same projective
closure, e.g. if e is the class of homom-_•rphic images of Q and
S) (see [351 or [41]).
In Chapter 3 we used implicitly the fact that if a is a torsion class (in ) and Cc To' then T( jve )
g n g-0' whence if a is closed under pure subgroups, the same
is true of T( LC). THEOREM 4.32 raises the question whether,
given a torsion class g, a torsion theory (a,$) and a subclass
e of U, it is possible for T(3 C) to contain objects of
which do not belong to J. By LEMMA 3.11, which obviously holds
also it may be assumed that e is a torsion class.
64.
PROPOSITION 4.33: Let (11„,) and (':1,a) be torsion theories for:k, the latter with idempotent radical r. If C", is a torsion subclass of ii., then U(05 = T(Uve )(-14 if and only if r(G) is ae -pure subobject of G for every G C S.
PROOF. An element K of T(ZTL)e)(1 belongs to a
if and only if K/r(K) = O. Since K/r(K) e T( 3 we.), this is
equivalent to s(K/r(K)) = 0, where s is the idempotent radical
fore- But s(K) = 0 so by COROLLARY 4.29, s(K/r(K)) = 0 if
and only if r(K) ise -pure in K. Thus if r(G) ise -pure in G
for every G c , then :j t\& = T( OU(3 )(1S . Conversely,
if this equality is satisfied, then any G C , r(G) is
the largest subobject belonging to T( Or u C )(.1L& , so
s(Glr(G)) =
The conditions of PROPOSITION 4.33 are not always
satisfied, as the following example shows.
EXAMPLE 4.34: For distinct primes q, t, consider the
torsion theory (T(Q({q,t})WS ) and the group G of EXAMPLE 4.25:
G = [p-n x, q-n y, t-n( y' x+ )In = 1,2,3,...]
where p is a prime other than q, t and x, y are linearly
independent. G , but since p G G, G T(Q(p))In`S .
However, there is a short exact sequence
0-4Q(p) [x] -- G -4G/[x] * Q({q,t}) -40
which shows that G belongs to T(T(Q(p)) U T(Q({q,0)))(1"1 .
Another problem suggested by results in Chapter 3
is that of determining when the class of extensions of objects
65.
in a torsion class a by members of a torsion class (0 2 is 1
itself a torsion class. (cf. PROPOSITION 3.16). The conditions
of PROPOSITION 4.33 are sufficient. Before showing this we prove
PROPOSITION 4.35: Let (3 g 1) and (a2' 9.2 ) be
torsion theories for :X. The following conditions on K e -3L
are equivalent.
(i) There exists a short exact sequence
--> K —4 K" 0
with K' e 1and K" 2'
(ii) There exists a short exact sequence
0 K K—) K"--> 0
with K' Ea and K" e rj 2 r1 (31 1 .
PROOF. Let (a1,) have idempotent radical rl.
(i) => (ii): There is a short exact sequence
0 —÷K'--> L —4 r 1 (K") 0
where LS. K and necessarily L e Z7 1 . The resulting exact sequence
O--)L --->Ku/r i (K") —40
satisfies (ii). Obviously (ii) => (1).//
PROPOSITION 4.36: Let (1 l' a1), (:72' c-j.2
) be
torsion theories for idempotent radicals r l , r2 . If
r2 (K) is :1 1 -pure in K, for every K e then the following
conditions are equivalent:
(i) K e T(J I U :1 2 )
(ii)and (iii) as in PROPOSITION 4.35.
PROOF. Clearly (ii) => (i) without restriction on
27 1 and 27 2 . By PROPOSITION 4.35 it suffices to show (i)
For every K C T( rcl i u a 2 ), there are short exact sequences
0 r (K) K K / r (K) = K" —?' O
and 0 r2 (K") -4 K"--4 K"/r2 (K") > 0,
the latter being J 1-pure. But then K"/r2 (K") c 2 n T611 u eCl2 ), so K"/r2 (K") = 0, i.e. K" e 2 rtZ,; 1 .11
4. Generalized Rank
For the remainder of this chapter we shall work in COG-Many results however can be generalized to modules over
hereditary rings (at least).
The torsion classes closed under subgroups and pure
subgroups have been classified by minimal representations. In
attempting a similar classification of torsion classes closed
under generalized pure subgroups, we therefore begin by
searching for groups which give simple representations of such
classes.
The hereditary torsion classes are determined by groups
of the form Z(p) and Z, which have no non-isomorphic proper
subgroups and for any torsion theory (g,a) belong to either
J or a. Groups of the form Q(P), Z(p) and Z(p) give
representations of all torsion classes closed under pure
subgroups, and a group of this kind has no proper pure subgroups
and belongs to a or c3' for any torsion theory (7,2). Some
information about torsion classes closed under e -pure subgroups might therefore come from the study of groups which belong to
66.
0 or a for any (Zf:3-) and which have no proper e -pure
subgroups.
When C is a torsion class, however, we can specify
groups determining torsion classes closed under e -pure subgroups
in terms of a generalized rank function which we now introduce.
To justify the definition of generalized rank the
following result is needed:
PROPOSITION 4.37: Let (11,;S) be a torsion theory.
IfG eS, then the intersection of any famag oft( -pure
subgroups of G is U -pure .
PROOF. By COROLLARY 4.29, it suffices to show that
G,a GA c '5 for any set {GA I X CA) of U -pure subgroups of G.
Suppose a GA C _c G and Gya GA cit . Then for each ti c A ,
we have a diagram
0 (G' n G )/ G —4 G'i rm G G'/(G' G XeA AEA 1.1
IR
67.
with exact rows
Gy x,A GA , G'/(
means that Cc_
0 _4(G'+% )/G P 11
, but by assumption on
Hence G'/(G I ( G) = 0. But this
G for each u, so V/21 GA = 0, i.e. GT, GA Cid/ 11
• (G'+G )/G c 11
G' ) e U .
Every element or subset of a group G eS is therefore
contained in a smallestlk,-pure subgroup.
The generalized rank for a torsion theory (a,S) is
introduced in the following definitions.
68.
DEFINITION 4.38: If u1 is a subset of the elements
of a group G [ LU ] t J.G denotes the smallestU-pure subgroup
of G containing W. If iu is a finite set {x1 ,... 9 xn} or a
countable set {x1 ,x2 ,x3 ,...}; [W) denoted by
or [x1 ,x2 ,...]u, respectively. When there is no confusion about
the containing group the superscript G will be omitted.
DEFINITION 4.39: A non -zero group G e5 has tt -rank
Ittif it has a subset 6 with [ 6 Ju. = G and I 5 I = ill and if
lytis the least cardinal number for which such a set exists. We
denote this by writing -U. -rank (G) = lii . 6 is called att -basis
for G.
If (2),S) = ( 0'g') this definition gives the 0
standard (torsion-free) rank, since in 2T- purity coincides with 0
10-purity. Note theta. -rank is defined only on non-zero groups
Obviously for every non-zero xe Get, [x] has
lt.-rank 1, so since G is generated by such subgroups, we have
PROPOSITION 4.40: If (tt,) is a torsion theory then
every G e \S is a homomorphic image of a direct sum of groups in
Using THEOREM 4.32 and PROPOSITION 4.40 and reasoning
as in the proof of THEOREM 3.9, we obtain
THEOREM 4.41: Let (7,a) and (1t , ) be torsion
theories such that :T is closed wider 7J -pure subgroups. Then •
= T(( n ) LJ {G e n ‘S I 1d4. -rank(G) =
69.
The groups with 2 0-rank 1 are the rational groups,
and we have seen that a torsion class is closed under 7 0-pure
subgroups if and only if it is closed under pure subgroups.
Also there is a rank function ({0}-rank) corresponding to the
trivial torsion theory ({0},0ir), with {D}-rank (A) = 1 if and
only if A is non-zero cyclic and cyclic groups determine the
hereditary torsion classes (closed under {0}-pure subgroups).
Thus THEOREM 4.41 is a generalization of THEOREM 1.34 and
THEOREM 3.9. Although the theory of types, divisibility and
algebraic compactness is too directly involved in the discussion
leading up to THEOREM 3.13 for any more detailed generalizations
to appear likely, the groups of generalized rank I nevertheless
seem to provide a convenient point of departure in the search
for representations for torsion classes with additional subgroup
closure properties.
An alternative description of generalized rank is
given by
PROPOSITION 4.42: G c"fl has a -rank m if and only
if it has a subset 6 with 1 5 1 =11-1, satisfying the following
equivalent conditions
(i) Gn 6 ]
(ii) If H e j and f:G --411 satisfies f(ti) = 0,
then f = 0,
and lii is the smallest such cardinal nuriber.
PROOF. We first verify the equivalence of (i) and
(ii). If (i) is satisfied and f:G H e.lb satisfies f(5) = 0,
70.
then there is a commutative diagram
where necessarily g = 0, so f = 0. If G/[6 ] 4LL , it has a
non-zero homomorphic image H in -5 . There results a non-zero
homomorphism G —3>G/ [ 5 ]-----3 H, whose kernel includes 5 .
Now let tfj be any subset of G with G = [ 1 11... . For
any non-zero epimorphism f:G/[ , f g( W ) = 0, where
g is the natural map G G/[ uj . But f g has all, -pure kernel,
which must contain [ Lti = G, i.e. fg = 0, so f = 0 and
G/ W . Conversely, if G/[ 113 ] e /4, , then G/[ W ] is
both a homoraorphic image of G/[ ] and a member of ss , i.e.
G/[ w = 0. Thus G = [Iii ] tx. exactly when G/[ W] cli. In
particular this is so when 11,1 is replaced by a set 6 of minimal
cardinality.//
COROLLARY 4.43: G e"sh has finite U. -rank n if and
only if there exist linearly independent elements x1 ,...,xn C G
with G/(x1 ) ® [xi) e11, and n is the least such integer.//
PROPOSITION 4.44: Let (111 ,31) and (11 2 ,) 2 ) be
torsion theories with 'j Then for any group G e 1 ,
<U,2-rank(G).
PROOF. Let6 be a1.1-2-basis for G. Then G/[5 ] e
c:1.11. , so 1.i J.-rank (G) <I 6 1.1/
71.
COROLLARY 4.45: If ,S) is a torsion theory with
then the 1.1. -rank of a group in S cannot exceed its
rank.//
COROLLARY 4.46: Let X be rational and ii the torsion
class {Al[A,X] = 0; then U -rank(X) = 1.
PROOF. D 0 c11, so
0 <11-rank(X) < rank(X) = 1.//
COROLLARY 4.47: For any torsion theory (11,S) with
U.; 0 :1A, -rank(G) <c0 -rank(G) for every group G C .//
One can introduce a notion of (LL,S)-independence in
groups III'S , which coincides with the standard linear independence
in torsion-free groups for (11,h) = C1 0 ,50): call an element
x of G (tt,9)-dependent on G if x E
It would be interesting to know of conditions on torsion theories
(forakror for module categories) under which this notion of
dependence gives an abstract dependence structure of the kind
studied by Kertesz [26], Dlab [10] and others. We shall not
study this question explicitly, though several examples of
pathology are to be found in the subsequent discussion.
For any torsion theory (11,,S), the class of groups
in‘ifj with finitelk, -rank is closed under homomorphic images in
and under extensions, but not under subgroups in general.
PROPOSITION 4.48: Let G e 'S have finite 1L -rank n. Then any
non-zero homomorphic image of G which belongs to 'S has ii. -rank
< n.
PROOF. Let {xl ,...,xn} be alt-basis for G,G/G' c
the set of distinct cosets of x l ,...,xn mod. G' and
G/G' = bri ,...,yr ],tk . Then GIG' islt -pure in GIG' and since
GIG' c G' istt, -pure in G. Since -IC-purity determines a A
proper class, G is thereforea-pure in G and since it contains A
xl ,...,xn , G = G, i.e. G/G' = i'''ri soll,t -rank (G/G')
<r <n./
COROLLARY 4.49: If G has L( -rank 1, so does any
of its non-zero homomorphic images in
PROPOSITION 4.50: If G° is I( -pure in G E , cald
if G I and GIG' have finite -U.-rank, so does G, and
-rank(G) < tt. -rank(G') +11..-rank(G/G')
(For an instance of strict inequality, see EXAMPLE 4.58 below).
PROOF. we chooselk-bases {x1 ''''' x } for G', n
fzi ,...,zml for GIG' and representatives y i for zi in G,
i = 1,...,m, and define groups
G = =
For any homomorphism c'S with g(Z) = 0, we have a
commutative diagram
) G ---g---4 K
where all other maps are inclusions. Since g k f' = g f h = 0,
we infer from PROPOSITION 4.42 that g k = 0 so there is a
homomorphism v:G/G'--4K such that vu = g, where u is the
72.
73.
natural map from G to GIG' and a commutative diagram
G'
f G ---IL--> K
(G4:G--)/C G/C
where f" is inclusion and t the natural map. From this,
v f"t=vuf=gf= 0, sovf" = 0, sincetis an epimorphism.
Also, (G'4Z)/G ° = [zi ,...,znI] so by PROPOSITION 4.42, v = 0.
Thus g = v u = 0, so as in the proof of PROPOSITION 4.42, we
have
solk-rank(G) <m + n =IL -rank(G') +1L-rank(G/G 1 ).//
The next example shows that a group with a generalized
rank I may have a subgroup for which the corresponding rank is
infinite.
EXAMPLE 4.51: Let p l , p2 , ... be the natural enumeration
of the primes, Y = Q(IPZni n
= 1,2,...1) and X = [ I In = 1,2,...]. P2n
Y hasSb -rank 1, since any element with zero height at all primes
p2n-1
gives alb -basis. Any finite subset Ix "" xm1 of X
a(n) generates a cyclic subgroup [x], and X/[x] 1 (E) Z(p ),n
n = 1,2,..., where a(n) E Z. Thus X/[x] ER„ so g -rank(X) = )ko .
5. Groups ofIt -rank 1
Among other things we wish to describe the groups G
of1A.-rank 1 ((1,T1) is a torsion theory throughout this section)
for which T(G) is closed under 71.-pure subgroups. This requires
in particular that G' 6 T(G) for everyll-pure subgroup G' of G.
The most obvious way in which this can be satisfied is for G to
have no propera-pure subgroups at all, and the possible
relevance of this property has already been noted (54). This
section is largely devoted to the structure of groups of -Li-rank
1. We begin by discussing groups G of 1J-rank 1 which have no
proper-pure subgroups and then consider G satisfying a
weaker condition: every propera-pure subgroup G' has finite
index. An example shows that even the weaker condition is not
universally satisfied and points to the difficulty in obtaining
an analogue of the type set to describe the groups of IL-rank 1
in a torsion class (cf.51 of Chapter 2). This difficulty is
also apparent in THEOREM 4.63, which partly generalizes
THEOREM 3.12.
The other main result on groups of -Lt.-rank 1 asserts
that they cannot be mixed.
DEFINITION 4.52: A group G is said to beli-pure
simple if it has no proper -Lk -pure subgroups.
PROPOSITION 4.53: A group G c U -ronk I is
IL-pure simple if and only if for every non-zero x E G, {x} is
alk-basis.
PROOF. If every non-zero element gives a 71-basis
and 0 G' G then for any non-zero x e G',
GIG' (G/ki)/(G'/[x])
74.
so G' is not ii-pure. Conversely, if G isli -pure simple
[x] = G for every non-zero x C
A torsion-free group is J 0-pure simple if and only
if it has J0-rank 1. It is not necessary that 11-rank(G) = 1
implies G istt-pure simple in general, although the last proof
shows that the converse implication always holds.
PROPOSITION 4.54: If for some prime p, Z(p)
then every U. -pure simple group in 4:j is either p-divisible or
isomorphic to Z(p).
PROOF. pG is 1J_,-pure in G for every G cj ., since
G/pG . Thus aIL -pt.:re simple group G e j5 which Ls not
p-divisible is p-elementary, and then necessarily cyclic.ft
Every non-trivial i5 contains Z which has tt.-basis
Ill. If for some prime p, Z(p) e"S , then by the previous result, Z is not11-pure simple. Thus we have
PROPOSITION 4.55: If every G e with a -raisk 1 is
lk -pare simple, then tCi
The converse of PROPOSITION 4.55 is false (cf. EXAMPLE
4.58 below).
In using groups withtt-rank 1 to classify torsion
classes closed underll-pure subgroups, the fact that such a
group G may have a properll-pure subgroup p G presents no
great difficulty, since we are essentially concerned only with
torsion-free G, and for such groups, p G G, so T(pG) = T(G) e
75.
We shall see in EXAMPLE 4.58 that much greater complexity of
11-pure subgroup structure of G is possible. First however,
we note a connection between single-element1L-bases and a
property which may be regarded as a generalization of 1k -pure -
simplicity.
PROPOSITION 4.56: IfU-rank(G) = 1, then the
following conditions are equivalent:
(i) For every non-zero x G and for every -U.-basis
fyl of G there exists a non-zero integer n such that n y e
(ii) Every proper1L-pure subgroup of G has finite
index.
PROOF. (i) => (ii): Let G ° be a proper-pure
subgroup of G, x e G ° , x 4 0, {5,} all-basis for G with
n y 6 [x] ci G ° , and H = [G ° ,y]. Then RIG' is cyclic with
order m = Zik > 0, k y e G'}. In the resulting short
exact sequence
0-42(m) 1. H/G 9 --4 G/H--* i1 we have HIV. -5 and GillEU. If m = p 1 prr , where
are primes, then RIG' = Z(p il)q) e Z(prr), and
(G/H)
is divisible for j = 1,2,...r, since no Z(p ) belongs Pi i 4
toll. If for some j, Z(pi") is embedded (in GIG') in a
co co subgroup isomorphic to Z(p ), then Z .) belongs to both 11, and (Pj
db, which is impossible. It follows thatZ J) = (G/G')
(R. for
) Pi
j = 1,2,...,r, so that H/G ° is a finite pure subgroup and hence
a direct summand of G/G ° (see for example [15] p.80). But
since G/G ° cand G/H cL , we then have H/G ° = G/G ° , so G °
76.
has finite index m.
(ii) => (i): If every properli-pure subgroup has
finite index and x 0, then either {x} is all-basis or [x] li.
has finite index, in which case for everyli-basis {y}, we have
n y e [x]11,
for some n e Z, n 0.//
COROLLARY 4.57: If G has -It -rank 1,is torsion-free
and satisfies the conditions of PROPOSITION 4.56 and if T(G)
is closed underll -pure subgroups, then T(G) = T(G') for every
proper -LI-pure subgroup G' of G.
PROOF. Since GIG' is finite, PROPOSITION 2.20 says
that G T(C). By assumption, G' e T(G).//
If every group withli-rank 1 satisfies the conditions
of PROPOSITION 4.56, and if in addition sic 3., then the groups 0'
with 'U-rank I are an IL-pure simple.
The next example shows among other things that the
conditions of PROPOSITION 4.56 need not be satisfied when
EXAMPLE 4.58: Let
TuQ(12,3))} {Z(p)all panes p})
and
G = [2-nx, 3-nyln = 1,2,...]
where x and y are linearly independent. Then Q(2) e Q(3) z
G c t. Denoting the cosets of x,y modulo (x+y] by ;,77
respectively, we have x and y with the same type, with infinite
2-height, 3-height respectively. Thus G/[207] e IL and G haa
li-rank 1,
77.
78.
Clearly everyli-basis of G must have the form {a+b}
with a c [ ], b E [y] * and a f 0, b 4 0, and a+b can have no
non-zero multiple in either [x],w = [x] * or [y] [y] * .
If a group of U-rank 1 is directly decomposable, then
every direct summand hasti-rank 1, by COROLLARY 4.49. Such a
group cannot be a direct sum of infinitely many non-zero
subgroups, for factoring out a cyclic subgroup leaves almost all
summands intact. It is however possible for direct products of
infinitely many groups to have generalized rank 1. Wiegold
[43] has shown, in effect, that 1-TI(p) has.-rank 1, where
the product is taken over all primes p. By COROLLARY 4.49 the
corresponding statement is true for any set of primes. For the
group G of EXAMPLE 4.58, we have T(G) = T({Q(2),Q(3)}), while
by PROPOSITION 3.3, T(Op 1(0) = Tai(p)lall p}). we are led
to ask whether, in general, the discussion of groups of 11-rank
1 in torsion classes can be reduced to consideration of
indecomposable groups. We must leave this question unanswered.
The conditions under which a rational group can have
generalized rank 1 are given by
PROPOSITION 4.59: Let X be a rational group belonging
to"5. 1A- -rank(X) = 1 if and only if T(x) = T(n1 ,n2 ,...) where co QD
ni = 0 if Z(pi) c , 0 or 00 if Z(pi ) E '5 but Z(p7)
PROOF. If T(X) is as described, let x c X have height n4
(n1 ,n2 ,...). Then X/[x] 61) Z(p i-), i = 1,2,... . If Z(p i) ni , then Z(pi ) cU.. If Z(pi) e 15 but Z(p7) , then I
00 (pi ./) = 0 or Z(p
i), and so belongs toll, , and if Z(p) e -5
79. n.
then Z(p1) = O. Thus X/[x] c'LL and Zt-rank(X) = 1. Conversely, if X has1),-basis {x}, then X/[x] is a direct sum of primary
groups, each belonging to U, and it is a simple matter to show
from the restriction this places on their orders, that the
height of x is as required.//
A non-zero direct summand of a group of IL-rank I
also basil-rank 1 (COROLLARY 4.49). Thus together with the
remarks following EXAMPLE 4.58, PROPOSITION 4.59 gives a
necessary condition for a completely decomposable group G c5 to have rUL-rank 1: G must have finite rank and the types of its
direct summands must be as in PROPOSITION 4.59. This condition
is not sufficient, however. For example let A and B be
isomorphic rational groups of V_-rank 1. If {x} is alt-basis
for ACE) B, then clearly A E) B/[x] * cll. But [x] * is a direct
summand (see for example [15113.166), so A e B/[x] * , which is rational, also belongs to . This clearly is impossible.
Although in the investigation of torsion classes only
torsion and torsion-free groups need be considered, there is
some interest in the fact that mixed groups cannot have
generalized rank 1. As a first step in showing this we prove
PROPOSITION 4.60: A torsion group with 71 -rank 1 is
cyclic.
PROOF. Let G c a 1)‘5 havelk -basis {x}. Then 0
since G/[x] has no more non-zero primary components than G, and
belongs tol)L, it follows that G/[x] is divisible and has zero
80.
00 p-component if Z(p ) j Taking primary decompositions of
Gt and [x], we have G/[x] = ( G )/((i) [x ]), where x = 0 P P P
for almost all values of p so that for such p, G = G ax ] is P P P
in both 45 and U, i.e. G = O. All the remaining G's are P P
reduced, (since factoring out a cyclic subgroup cannot eliminate
a non-zero divisible subgroup) and have G /[x ] divisible. If P P
x has finite height (in G) it is contained in a cyclic direct
summand (see for example [15] p.80), whose complementary
summand must vanish as it is divisible. There remains the case
where x has infinite height. In this case, if y E G, then
for any positive integer n, there exists y' e G such that
y - pny , = m x, for some m E Z, since G /[x ] is divisible. P P P
But then x = pnx, for some x' E C, whence it follows that y
P P
has infinite height and G is divisible. With this contradiction P
the proof is complete.//
PROPOSITION 4.61: A group withlt-rank 1 is either
torsion or torsion-free.
PROOF. Let G E "S be mixed and have all-basis' tzl.
If x e Gt' then Gt /[x] is pure in (G/[x])t' so by THEOREM 1.38
and PROPOSITION 3.7 G/[x] c a. But then11.-rank(G t ) = 1, so
by PROPOSITION 4.60, Gt is cyclic which means that G splits
(see for example [15] p.80) and this is not possible since a
summand complementary to G t is not affected by the factoring
out of [x]. Thus x must have infinite order. As in the proof
of PROPOSITION 4.60 it can be seen that Gt is reduced whence it
follows that G has a direct summand of the form Z(p 1 ) ([15] p.80).
Let such a summand be generated by y. Then x = k y + z, where
k e Z, k y f 0 and z has infinite order. Since u contains no reduced p-groups, [y]C [x], i.e. for somemeZ, y= mx =mky
+ in z. But then linear independence of y and z requires that
mz= 0, i.e.m=0andy=mky= O. Again we have a
contradiction.//
The next theorem is a partial analogue of THEOREM 3.12.
Its proof makes use of the following lemma.
LEMMA 4.62: If G ° is -U. -pure in G e t then
G' = [x] for fbr any x e G'.
PROOF. [x] G' ij is, -pure in G, so [x] G C[x] G, •
But x e [xi r\ G' which -pure in G', so [x] G' C [x] G r‘G / ,G
1k
THEOREM 4.63: Let C c:S be a class of groups with
11 -rank 1, 652 the. class ofhomomorphic images in %5 of direct sums of copies of groups in (2 , r, s the idempotent radicals for
T((L), Ife is closed under extensions and satisfies
C1 ,...,Cn c C , C ij -pure in C l e ... cn ,
(c) = 1 => C e CI
then T((;) is closed under 11.. -puresubgroups if and only if s r
is idempotent.
PROOF. For a group A , we define a subgroup A
to be generated by all elements of A which belong to the images
of homomorphisms from groups C e C . Clearly A e TO for any
81.
If (A/K) = A'/K, we have an exact sequence
0 ->A -4 A° -4 (A/A) ---> 0
whence A° e C, so A = T, (ArA) = 0 and A = r(A). In other
words, ve ) = e. Thus for any A e T(C ) , with a -
pure subgroup A' we have an exact diagram
0 —fl3 ED C A —4 0 AEA
A' K/B
0
where each CA c (! and K is the inverse image of A' in EE) c.
B isIL-pure in (I) CA, since A e‘'.5 ; also K/B 1st" -pure in .
(i)CA/B. Therefore K is It-pure in ecx . For any y C K, we
have, after suitably re-labelling, y C C 1 •.. E) Cn , which
is 10,-pure in CA , so [y]K
= [y] is lk-pure in
C1 e Ept Cn and therefore belongs to C . Thus K .c T(f),
so A' c T(e ). T(e) is therefore closed under 1L-pure
subgroups. The result now follows from THEOREM 4.32.//
6. Groups of -U.-rank 1 (continued)
We commence this section by characterizing the groups
of cip -rank 1, where p is a prime. As a first step we prove
PROPOSITION 4.64: Let G be p- reduced with -rank 1.'
Then for any Sf)_basis {x} of G, [x] is a p-pure subgroup.
PROOF. It clearly suffices to show that G/[x] has no
direct summand of the form Z(p c°). By PROPOSITION 4.61 it may be
assumed that G is torsion or torsion-free, and in the former
case G = [x], by PROPOSITION 4.60. There remains only the
82.
torsion-free case. If G/[x] has a summand Z(p ) then G has a
subgroup G', containing x, for which the sequence
0 —4. [x] -->G ° Z(p°Q) —4 0
is-exact. But then G ° Q(p),([15], p.149) contrary to the
assumption that G is p-reduced.//
This result will enable us to give a complete
description of the groups with o0 -rank 1. We first recall a
definition introduced by Fuchs [161:
DEFINITION 4.65: A subgroup B of a group A is called
a p-basic subgroup, where p is a prime, if B is a direct sum of
cyclic groups of infinite and/or p-power order, B is p-pure in
A and A/B is p-divisible.
We have shown in PROPOSITION 4.64 that if G e 64., has
a -basis' {x}, then [x] is a p-basic subgroup. On the other
hand, if a p-reduced group has a cyclic p-basic subgroup [y],
then {y} is clearly a .9 -basis.
PROPOSITION 4.66: If -rank(G) = 1 and x G, then
{x} is a -basis if and only if DO is a p-basic subgroup.//
A p-reduced torsion group must be a p-group and it is
shown in [3] that a torsion-free p-reduced group has a cyclic
p-basic subgroup if and only if it is isomorphic to a p-pure
subgroup of I(p). These observations, with PROPOSITION 4.66,
give a proof of
83.
84.
THEOREM 4.67: A group G p has SO -rank 1 if and only if it is isomorphic to either a non-zero p-pure subgroup of
l(p) or Z(pn) for some finite n.//
If a torsion group G has -rank 1, only a generator
can give a 9 -basis. In the case of torsion-free G; the 4) - P
bases are described by
PROPOSITION 4.68: Let G be a torsion-free, p-reduced
group with)-rank 1, viewed as a p-pure subgroup of 1(p).
The following conditions are equivalent for x e G:
(i) x has p-height 0 (in G and hence in I(p)).
(ii) x is a p-adic unit.
(iii) {x} is as-basis for G.
PROOF. The equivalence of (i) and (ii) is well-known.
By a theorem in [2], G/[x] is p-divisible if and only if [x]
contains a p-adic unit, and this is so precisely when x itself
is a unit.//
Thus when G is torsion-free of ciD -rank 1, every non-
zero element has the form pnx, where {x} is a 00 -basis.
The results obtained thus far enable us to give a
rough description of the groups of.4516-rank 1, where S is a set
of primes. In the uninteresting torsion case they are the
cyclic S-groups; in the torsion-free case they are described
by
0 —3 f] = ([x] + 0 1 (p))/(0'(p))--> G G t (p)--3G/([x]+G ° (P))--.40.
0
PROPOSITION 4.69: A non-zero S-reduced, torsion-free
group G hasog s-rank 1 if and only if
(i) for each p c S there is an exact sequence
0 -4 G(p) -3 G -4 0"(p) 0
where G t (p) is p-divisible and G"(p) is isomorphic to a p-pure
subgroup of I(p), and
(ii) there exists an element x of G such that for
every p E S with pG G, the image of x in G"(p) has p-height O.
PROOF. Lets-rank(G) = 1 and let {x} be a 5-basis
for G. For p e S such that G is p-reduced, {x} is as -basis,
since G/[x] is S-divisible and hence p-divisible. Thus G is
isomorphic to a p-pure subgroup of I(p) and we let G t (p) = 0,
G"(p) = G. If G is p-divisible, we let G(p) = G and G"(p) = O.
Finally we consider p c S such that G is neither p-divisible nor
p-reduced. Let G e (p) be the maximal p-divisible subgroups of G
and G"(p) = G/G t (p). Denoting the coset of x mod.G t (p) by -X:,
we have a commutative, exact diagram
0 [x] > G/ [x] >0
85.
G/[x] is S-divisible and hence p-divisible, so the same is true
of Gi([x] + G t (P)). It therefore follows that G/G t (p) = G"(n)
has -rank 1 and thus has a p-pure embedding in I(p). As
defined, G"(p) 0 exactly when pG G and in such cases it is
clear from the proof so far that the image of x in G"(p) has
p-height O.
86.
Conversely, if G satisfies (i) and (ii), we shall show
that for x as described in (ii), x is a 25-basis. If G"(p) = G,
then x has p-height 0, so GI[x] is p-divisible and if G"(p) = 0,
G itself is p-divisible. In the remaining case, G"(p)/(R] is
p-divisible, where again -; denotes the image of x in G"(p). But
G"(p)/[5];-(G/G c (p))/(axJ+G'(P))/G'(P))G/Ux]+G'(p))
(G c (p) has been treated as a subgroup of G), and ((x]i-G c (p))/(xj
is also p-divisible, so from the exact sequence
0 —4 ([x]+G'(P))/[x]--)G/M-->G/([x]+G' (P)) ----> 0
it then follows that G/[10 is p-divisible. Thus G/[x] is
p-divisible for every p e S, so {x} is a 019) 5-basis and the proof
is complete.//
If the groups G c (p) are regarded as subgroups of G,
they are all pure, so their intersection is also pure in G and
hence in each G c (p). Being therefore S-divisible, G c (p) = O.
Thus we have
COROLLARY 4.70: A torsion free group with j,N-rank 1
is a subdirect product of torsion-free groups with 2 p-rank 1,
at most one for each p e S.//
In our discussion of groups of generalized rank 1, we
have not so far examined the question when such groups G
satisfy
(*) G e D* or 3. for any torsion theory (L3).
(This condition is satisfied by groups of a ct-rank 1). PROPOSITION
4.69 provides a characterization of the groups of-rank 1
which satisfy (*):
87.
PROPOSITION 4.71: With the notation of PROPOSITION
4.69, and s the set of all primes, if G is torsion-free and
jp-rank(G) = 1, then G satisfies (*) if and only if for every
p, G = G t (p) or 0(p).
PROOF. By definition of the groups G'(p) and G"(p),
the stated condition is necessary for (*). Conversely, if G
satisfies this condition it is cohesive in the sense of [11],
and so G/r(G) is divisible for any idempotent radical r with
r(G) 4 0 (r(G) is a pure subgroup). Thus if r(G) 4 0, r is
associated with a torsion class containing torsion-free groups,
so that G/r(G) = r(G/r(G)) = 0, i.e. G = r(G).//
There are groups of 4 -rank 1 which do not satisfy
(*), e.g. TrI(p), p c S if S has at least two elements.
To obtain a description of the class of groups of
1-rank 1 in a torsion class :I, analogous to that of the type set given by THEOREM 2.11, it is first necessary to find
conditions on groups A,B withlk-rank(A) = lk-rank(B) = 1 which
ensure that B E T(A). For a really close analogy with §1 of
Chapter 2, we should restrict our attention to those torsion
theories (11,) for which all non-zero homomorphisms between
groups withli-rank 1 are monomorphisms. Such is the case when
the groups with/L.-rank 1 are all /1-pure simple, though as we
shall see in the case of (29 a.) this condition is not P' P
necessary. For such a (71,11) let A and B have 1L-rank 1, with
[A,B] O. Then we may assume that Ac B, and it follows that
B belongs to T(A) if and only if B/A does. If (10) = ejar:T0),
88.
then in every case B C T(A) and B/A V o . If every group with
1L-rank 1 is IL-pure simple, then B/A C 'ft always, but EXAMPLE 4.34 raises doubts as to whether B c T(A), though certainly
B e T({AJ L/C) for some subclass e 001, and in particular
whene =
A complete set of conditions under which B c T({A} ulj,)
may be regarded as a partial generalization of THEOREM 2.11,
since in §1 of Chapter 2 the class T(X), X rational, could as
well have been replaced by T({X} U C) for any Cl C: rjo , as the
inclusion of extra torsion groups in a torsion class zr does
not enlarge jn g. o . Similarly, if j has type set r , so does T( u C.) if
We shall investigate the question for (U, ) =
where p is a prime. We begin with a description of
homomorphic images of groups with S -rank 1.
PROPOSITION 4.72: Let B be torsion-free of -rcnk
1. Then any proper homa;ro-Thic image of B is the direct sum
of a p-divisible group and a bounded p-group.
PROOF. If 0 +ACB andn= min {p-height inB of
ala e A}, then A C pnB and if a C A has p-height n in B, it has
n n+1 , zero p-height in p since otherwise a = p b for some b E B.
Being isomorphic to B, pnB has -rank 1, so by PROPOSITION
4.68, {a} is a p-basis for pnB. Since pnB/[a] belongs to
, so does PnBili, and thus Ext(B/pnB, pnB/A) = 0, so the
natural exact sequence
0---> pnB/A B/pnB 0
is split,//
89.
COROLLARY 4.73: If B is torsion-free with.,) -rank1,
then B/A is a bounded p-group for any properA-pure subgroup
A./1.
COROLLARY 4.74: If A, B are torsion-free with QZ -rank
1, Men any non-zero homomorphism f:A-413 is a monomoyphism.
PROOF. If f has non-zero kernel, its image is both a
subgroup of B and the direct sum of a p-divisible group and a
bounded p-group, so f = 0.//
If AS; B and both groups are torsion-free with
JD -rank 1, then since T(A) contains all p-groups, we have B c T({A} UcEl ). This fact with COROLLARY 4.74 gives
PROPOSITION 4.75: The following conditions are
equivalent for torsion-free groups A and B with) -rank 1:
(i) B C T({A} ti)
(ii) [A,B] 0
(iii) A is isomorphic to a subgroup of B.//
I(p) has oe -rank 1 and has subgroups isomorphic to
all other torsion-free groups with SO -rank 1. In the case of
70-rank, Q plays a similar role. A further similarity between
the two groups is noted in the following proposition (cf. COROLLARY 2.5).
PROPOSITION 4.76: The following conditions are
equivalent for a torsion class
(i) g contains a non-zero torsion-free p-reduced
group.
90.
(ii) g contains a non-zero torsion-free group which
is not p-divisible.
(iii) 1(p) e
PROOF. Obviously (i) => (ii) and (iii) => (i);
(ii) => (iii) is just LEMMA 3.1.//
To conclude the discussion of groups with generalized
rank 1 we describe some torsion classes closed under -U.-pure
subgroups when U is determined by groups Z(pc°) (for various
primes p).
PROPOSITION 4. 77 : A group has (ap ncO)-rank 1 if
and only if it is either a non-zero cyclic torsion group or
isomorphic to Q(S), where S C P.
PROOF. Clearly the groups indicated have (Up n )-
rank 1, and for the converse, by PROPOSITIONS 4.60 and 4.61, we
need only consider torsion-free groups. If such a group G has
(U AcZ ) -rank 1, there is an exact sequence
0 Z G-4 G" —4 0
with GI ' e nZ. Thus G has rank 1 and G" = G" p C P
where G" 0 or Z(pc°). It follows that G Q(S), where
S =fp e PIG"p:= Z(pc°)) .11
THEOREM 4.78 : If ( gp A ) -rank(G) = 1, then T(G)
is closed under (U p ng)) -pure subgroups.
PROOF. If G is a torsion group, then T(G) is hereditary;
if not, then T(G) =0 5' 9 where S C P. The idempotent radical
91.
of Zi n S commutes with all others (cf. the proof of PROPOSITION
4.19) so in this case, by THEOREM 4.32, it suffices to consider
S-divisible groups with no direct summands Z(p ), for p E P. If
* * ) 0 —)A'---->A -->A" —40
is exact, with A,A"
without direct summands Z(p ), p c P,
then A" = 0• But then (**) is S-pure, so A' E S
direct summand Z(p ), p c P.//
and has no
7. An Example
To conclude this chapter, we find necessary and
sufficient conditions on rational groups X and Y for the closure
of T(X) under T(Y)-pure subgroups.
Our notation for heights, types etc. largely conforms
to that of [15], Chapter VII. In particular, p 19 p2 ,... is the
natural enumeration of the primes, and in a height (h1 ,h2 ,...,
hn h
n denotes height at p
n.
THEOREM 4.79: Let X,Y be rational such that T(X) is
the type of a height (h 1 ,h2 ,...,hn ,...) with 0 <h < co for
infinitely many values of n. Then T(X) is closed underT(Y) -pure
subgroups if and only if T(Y) < T(X).
PROOF. Let (T(Y),3, ) be the torsion theory for T(Y)
and let (g1 ,g2 ,...,gn ,...) be a height with the same type as Y.
If T(Y) < T(X), then X E T(Y), so T(Y)-pure subgroups
are T(X)-pure. For groups in T(X), such subgroups are direct
summands, and so belong to T(X) themselves.
For the converse we need to consider two cases:
(i) T(Y) T(111+1,h2+1,...,hn+1,...). Let M =
{flIhn = 00}. Let (k1 ,k2 ,...,kn ,...) be the subsequence of
positive finite terns of (h 1 ,h2 ,...,hn ,...) and re-label the
associated primes as q 1 ,q2 ,... . Let {x,y} be a basis for a
2-dimensional rational vector space and -k -k
G nx,qn n (cfr-111.0.0 1 p e m, n 1,2,...3.
A routine argument using the linear independence of x and y
shows that x has height (h 1 ,h2 ,...,hn ,...) in G. Suppose y is
divisible by qn n for some n. Since the same is true of
-1 qn x+y, x has qn-height kn+1 at least, which is impossible.
Thus T(y) < T(x) = T(X) (in G). Denoting the coset of y mod.
[x] * by 3T, we have -k
— — G/[x] * = [p-n qn n YIP e M, n =
so G/[x]* is rational with type T(X). From the exact sequence
0 --) X Z: [4 * -- ) G [x] *
it is clear that G E T(X).
Observing that [y] ic it T(X), we now show that [y] * is
T(Y)-pure in G. Let denote the coset of x mod. [y] *. Then -(k
n41)
G/[y] * = [p-112
qn Yelp E M, n = 1,2,...
which is rational of type < T(hi + 1, h2 + 1, ...,hn + 1, ...),
so C/[y] * c 3, and [y] * is T(Y)-pure in G.
(ii) T(Y) < T(h1+1,h2+1,...,hn+1,...). Let
U = r ipn lpn Y = Y1 and S = {pn Ihn < gn 1.
Note that our assumption concerning T(Y) requires that p X = X
for all p e U, S is infinite and g n is finite for each pn C S.
92.
93.
Let
V = fpn Ihn > gn Ign <col.
and re-label the entries of (h1 ,h2 hn,...) as follows:
denote the primes pn 6 S by s1 ,s2 ,..., their heights by k 1 ,k2 ,...
and denote the primes in V by v 1 ,v2 , with heights j l ,j 2 ,.•• .
Finally let
-k -j-jn -n -k -j -jn -k2n -1 H = [p
-nx, sn x, vn x, p y, 5 2n_, y, vn y, s2n (sZnx")
1p e U, n = 1,2,...].
As in case (1), T(y) < T(x) = T(X), [x] * ; X ; H/[x] * and
H c T(X). Also,
-k -(k -(k +I) -j -n_ x, s2n v 2n n_i = [p x' 52n-1 xip
U, n = 1,2,...]
which is rational with type T(Y), since it has lower height
at infinitely many privies, namely s 2n_1 , n = 1,2,... . Hence
[y] * is T(Y)-pure in H, but [y] *
THEOREM 4.79 has some obvious minor generalizations:
If T(Y) is replaced by an r.t. torsion class whose type set has
a least element, the theorem remains true. If Zris an r.t.
torsion class whose type set r has a subset r , of minimal
elements such that for every y e r there is y' e r' with y > y'
and if in addition T(h1+1,h2+1,...,hn+1,...) for every
y' e rl, then the argument in case (i) of the proof of THEOREM
4.79 can be easily modified to show that T(X) is not closed
under a-pure subgroups.
Only the case X ; Q(P) now remains. Here we prove
a more general result.
94.
THEOREM 4.80: = T(Q(P)) is closed undera -pure
subgroups, for a torsion class U, if and only if Z(pc°) c a
for every p e P.
co PROOF. If Z(p ) C a for each p E P, then by
PROPOSITION 4.77 and THEOREM 4.78,-Z p is closed under
( r4n6;1191)-pure subgroups and in particular,a -pure subgroups.
Conversely, if Z(Pc°) g for some p C P, then ,27 is co
a t-torsion class, and Z(p ) e 2p, where (a,g) is the torsion
theory associated with a. The natural exact sequence
0 --)Q(PL.{p}) Q(P)--> Z(p) —) 0
is accordingly 0. -pure, but Q(P-{p}) ftcl%,•//
COROLLARY 4.81: If a is not a t- torsion claw, (in
particular if g = T(y) for some rational Y), then) p is
closed undera -pure subgroups.//
95.
CHAPTER 5
MISCELLANEOUS TOPICS
1. Rational Groups and the Amitsur Construction
The Amitsur radical construction described in Chapter
1 does not always terminate after a finite number of steps, even
in abelian categories. If for example we begin with the class
(Z(p)) where p is prime, then for any group G, we have G n = G[pn ],
so in this case there is no finite upper bound on the number of
steps which may be required. In this section we shall discuss
the Amitsur construction for the idempotent radical r corresponding
to T(X), where X is rational, starting from 00. Thus for any
group G, we define G1 to be the subgroup generated by the images
of all homomorphisns from X to G, and G = G or (G/Ga-1
) 1 E3 a 13 a G /G
8-1 according as a is a limit ordinal or not. a
PROPOSITION 511: If G is a torsion group, then
r(G) = G1 for every rational group X.
PROOF. For any prime p with pX X, let y c G have
order pn ; then X/pnX [y] so Gp C: Gl . If pX = X, let Gp =
D E) R, where D is divisible, R reduced. Then D G1 but
[X,R] = 0. It follows that G1 = CE)G (P) , where G(P) is G if
pX X and otherwise its divisible part. This clearly is r(G).//
PROPOSITION 5.2: If X = Q(P) for some set P of primes,
then r(G) = G1 for every group G.
PROOF. Since r(G) is the maximum P-divisible subgroup,
its maximum P-subgroup is divisible. In a complementary direct
summand H of r(G), divisibility by powers of primes in P is
uniquely defined, so H, as a Q(P)-module, is a homomorphic image
of a direct sum of copies of X. Hence r(G)51 G i , so the two
subgroups coincide.//
THEOREM 5.3: If X = Q(P) for some set P of primes,
then r(G) = G1 for all torsion-free groups G. Otherwise there
exists, for each positive integer k, a torsion-free group G (k)
of rank k such that
r(G(k)
) = G(k)
= G1 ' .
PROOF. PROOF. Only the case X Q(P) needs to be considered.
Let T(X) = a and let (hl' h2" ' h ...) be a height of type a, n
the subsequence of finite non-zero terms of
(h19 h2 ,...,hn ,...), q1 ,q2 ,...,qn ,... the associated primes. The
groups G(k)
are defined by
-n -in -in -1
G (k) = [pxl , p-n
xk , qn xl , qn (qn xl + x2), ...,
-in -1 qn (cmn xk_i+ xk)IpX = X, (p prime), n = 1,2,...]
where {x1 ,x2 ,...,xn ,...} is a linearly independent set.
We first show, by induction, that
(k) _
G1 - [xl ] * , k = 1,2,... .
Note that A1 = A(a) for any torsion-free group A.
Now
G(1)
=
[p x1, q n = X, n = 1,2,...] ; X
96.
97.
so G (1) (a) = G(1) = [xl ] * . Assume G 1 (o) = [x1 ] * . Denoting
the coset of an element x by x, we have
-jn_n -1- r _ G
(k)/[xl ] * = L -n- -n
p x2 ..... p xk , qn x2 , qn (qn x24.55), ...,
-in -1— % (qn xk-1+&) IPX = X, n = 1,2,...
is thus isomorphic, in an obvious way, to
For any y e G(k) (a), TT' belongs to ri2 4, so y belongs to [x 1 ,x2 ] * .
Let my = m1x1+m2x2 where m,m1 ,m2 e Z. From the definition of
G (k) it is clear that x1 has height (h1 ,h2 ,...,hn ,...) and
in therefore type a. Suppose x2 is divisible by qn for some n. jn+1 Since the same is true of
qn1x1+x2' x1 is divisible by qn
and this is impossible. Thus T(x 2 ) < a. But m2x2 =
and T(y), t(x 1) > 0, so m2 = 0 and y e [x1 ] * . This proves the
assertion.
Now for any k, again denoting the coset of x by x, we
have (G(k) /[x1 ] *)(a) = [lc2 ] *, so from the exact sequence
(k) (k) (k) (k) 0-4 G(k)
(a) = -4 G2 -4 (G /G1 ) 1 = -30
(k) we deduce that G2
= [xx2
]*
G (2) and repetitions of this
argument give an ascending series
(k) c: G(k) c: c: (k) 1 * 2 * Gk
N 11? II
G(1)
G(k) G
(2)
so that r(G(k)
) = G(k)
.//
By taking k = 2 in THEOREM 5.3, we obtain
COROLLARY 5.4: (G/G(a))(a) = 0 for all torsion-free
98.
groups G if and only if a = T(Q(P)) for some set P of primes.//
The groups G(k)
of THEOREM 5.3 are indecomposable.
For suppose G H OD K; then G e T(X) and (k)
X = G (G) = H(a) @ K(a)
so H(a) = 0 or K(a) = O. But since H and K both belong to T(X),
this means that one of them must be zero. In addition,
X (fork >1). Thus (cf. Chapter 2) we have proved
COROLLARY 5.5: If a rational group X does not have the
form Q(P) and if k is any positive integer, there exists an
indecomposable torsion-free group A of rank k such that
T(A) = T(X).//
2. Splitting Idempotent Radicals
In Chapter 2 we saw that r(A) is a pure subgroup of A
for every group A and idempotent radical r. The cases in which
r(A) is always a direct summand are described by the next
proposition.
PROPOSITION 5.6: Let (7,7) be a torsion theory for (113.
with idempotent radical r such that r(A) is a direct summand
for every A. If r is non-trivial, then OS £) (and thus r(A) is always the divisible subgroup or its S-component for some
fixed set S of primes).
PROOF. If Z(p) c a for every prime p, then all groups in a are divisible. If for some prime p, Z(p) belongs to g , then so do all p-groups. If contains non-zero groups,
99.
it contains in particular TT Zn, n = 1,2,..., where Zn .; Z.
A theorem of Baer [4], Erdn's [13] and Saisiada (see [15] p.190)
asserts that Ext(T-TZn ,G) = 0 for a p-group G if and only if G
is the direct sum of a bounded group and a divisible group. Let
A' be a reduced, unbounded p-group and consider a non-split short
exact sequence
0-4 A ° TrZn -4 O.
Since A' E a and I-Tzn c 2F, we have r(A)
INDEX
determined (a torsion class by an object or class) 3.
generalized purity 48 et seqq.; see also 45.
generalized rank 68.
minimal representation 14.
n -free 27.
P-divisible 11.
P-group 11.
P-reduced 11.
p-divisible 11.
p-pure 45.
p-reduced 11.
r.t. torsion class 19.
S-pure 45.
T(r,P) 19.
t-torsion class 13.
type set (of torsion class) 19.
lk -basis 68.
.A,-pure simple 74.
1k-rank 68.
Containing new or uncommon terms and symbols only. See also the table of notation on pp. (ix)-(x).
100.
101.
REFERENCES
1. AMITSUR, S.A.: A general theory of radicals II. Radicals in rings and bicategories. Amer. J. Math. 76(1954) 100-125.
2. ARMSTRONG, J.W.: On p-pure subgroups of the p-adic integers. pp. 315-321 of Topics in Abelian Groups. Chicago: Scott, Foresman and Company, 1963.
3. : On the indecomposability of torsion-free abelian groups. Proc. Amer. Math. Soc. 16(1965) 323-325.
4. BAER, R.: Die Torsionsuntergruppe einer abelschen Gruppe. Math. Ann. 135 (1958) 219-234.
5. CHARLES, B.: Sous-groupes fonctoriels et topologies. pp. 75-92 of Studies on Abelian Groups. Paris: Dunod; Berlin-Heidelberg-New York: Springer-Verlag, 1968.
6. CHASE, S.: On group extensions and a problem of J.H.C. Whitehead. pp. 173-193 of Topics in Abelian Groups. Chicago: Scott, Foresman and Company, 1963.
7. CORNER, A.L.S.: A note on rank and direct decompositions of torsion-free abelian groups II. Proc. Cambridge PhiZos. Soc. 66 (1969) 239-240.
8. DICKSON, S.E.: On torsion classes of abelian groups. J. Math. Soc. Japan 17 (1965) 30-35.
9. . A torsion theory for abelian categories. Trans. Amer. Math. Soc. 121 (1966) 223-235.
10. DLAB, V.: General algebraic dependence relations. Publ. Math. Debrecen 9 (1962) 324-355.
11. DUBOIS, D.W.: Cohesive groups and p-adic integers. Publ. Math. Debrecen 12 (1965) 51-58.
12. ERDOS, J.: On direct decompositions of torsion free abelian groups. Publ. Math. Debrecen 3 (1954) 281-288.
13. : On the splitting problem of mixed abelian groups. Publ. Math. Debrecen 5 (1958) 364-377.
14. FUCHS, L.: On generalized pure subgroups of abelian groups. Ann. Univ. Sci. Budapest. E6tva Sect. Math. 1 (1958) 41-47.
15. . Abelian Groups. Budapest: Akadgmiai Kiad6, 1958; Oxford: Pergamon Press, 1960.
102.
16. FUCHS, L. : Notes on abelian groups II. Acta Math. Acad. Sci. Hungar. 11 (1960) 117-125.
17. . . . Recent results and problems on abelian groups. pp. 9-40 of Topics in Abelian Groups. Chicago: Scott, Foresman and Company, 1963.
18. GARDNER, B.J.: Torsion classes and pure subgroups. Pacific J. Math. 33(1970) 109-116.
19. . A note on types. Bull. Austral. Math. Soc. 2 (1970) 275-276.
20. de GROOT, J.: Equivalent abelian groups. Canad. J. Math. 9 (1957) 291-297.
21. Indecomposable abelian groups. Nederl. Akad. Wetensch. Proc. Ser. A 60 (1957) 137-145.
22. HEINICKE, A.G.: A note on lower radical constructions for associative rings. Canad. Math. Bull. 11 (1968) 23-30.
23. HULANICKI, A.: The structure of the factor group of the unrestricted sum by the restricted sum of abelian groups. Bull. Acad. Polon. Sci. Sr. Sci. Math. Astronom. Phys. 10 (1962) 77-80.
24. JANS, J.P.: Binge and Homology. New York: Holt, Rinehart and Winston, 1964.
25. JONSSON, B.: On direct decompositions of torsionfree abelian groups. Math. Scand. 5 (1957) 230-235.
26. KERTiSZ, A.: On independent sets of elements in algebra. Acta Sci. Math. (Szeged) 21 (1960) 260 -269.
27. KUROSH, A.G.: Radicals in rings and algebras. hat. Sb. 33 (1953) 13-26 (Russian).
28. . Radicals in group theory. Sibirsk. Mat. Zhurnal 3 (1962) 912-931 (Russian).
29. : Radicaux en thgorie des groupes. Bull. Soc. Math. Belg. 14 (1962) 307-310.
30. LEAVITT, W.G.: Sets of radical classes. Publ. Math. Debrecen 14 (1967) 321-324.
31. MAC LANE, S.: Homology. Berlin-Gatingen-Heidelberg: Springer-Verlag, 1963.
32. MARANDA, J.-M.: Injective structures. Trans. Amer. Math. Soc. 110 (1964) 98-135.
103.
33. MITCHELL, B.: Theory of Categories. New York-London: Academic Press, 1965.
34. NUNKE, R.J.: Slender groups. Acta Sci. Math. (Szeged) 23 (1962) 67-73.
35. RICHMAN, F., C. WALKER and E.A. WALKER: Projective classes of abelian groups. pp. 335-343 of Studies on Abelian Groups. Paris: Dunod; Berlin-Heidelberg-New York: Springer-Verlag, 1968.
36. S4SIADA, E.: Proof that every countable and reduced torsion-free abelian group is slender. Bull. Acad. Polon. Sci. Sear. Sci. Math. Astronom. Phys. 7 (1959) 143-144.
37. SERRE, J.-P.: Groupes d'homotopie et classes de groupes abelians. Ann. of Math. 58 (1953) 258-294.
38. SHCHUKIN, K.K.: On the theory of radicals in groups. Sibirsk. Mat. Zhurnal 3 (1962) 932-947 (Russian).
39. SHUL ° GEIFER, E.G.: On the general theory of radicals in categories. Amer. Math. Soc. Transl. (Second Series) 59, 150-162. Russian original: Mat. Sb. 51 (1960) 487-500.
40. SULINSKI, A., R. ANDERSON and N. DIVINSKY: Lower radical properties for associative and alternative rings. J.London Math. Soc. 41 (1966) 417-424.
41. WALKER, C.P.: Relative homological algebra and abelian groups. Illinois J. Math. 10 (1966) 186-209.
42. WALKER, E.A.: Quotient categories and quasi-isomorphisms of abelian groups. pp. 147-162 of Proceedings of the Colloquium on Abelian Groups, Tihany, 1963. Budapest: Akad4miai KiadO, 1964.
43. WIEGOLD, J.: Ext(Q,Z) is the additive group of real numbers. Bull. Austral. Math. Soc. 1 (1969) 341-343.